In the previous two quarters, we have been focusing largely on the “soft” side of real analysis, which is primarily concerned with “qualitative” properties such as convergence, compactness, measurability, and so forth. In contrast, we will begin this quarter with more of an emphasis on the “hard” side of real analysis, in which we study estimates and upper and lower bounds of various quantities, such as norms of functions or operators. (Of course, the two sides of analysis are closely connected to each other; an understanding of both sides and their interrelationships, are needed in order to get the broadest and most complete perspective for this subject.)
One basic tool in hard analysis is that of interpolation, which allows one to start with a hypothesis of two (or more) “upper bound” estimates, e.g. and , and conclude a family of intermediate estimates (or maybe , where is a constant) for any choice of parameter . Of course, interpolation is not a magic wand; one needs various hypotheses (e.g. linearity, sublinearity, convexity, or complexifiability) on in order for interpolation methods to be applicable. Nevertheless, these techniques are available for many important classes of problems, most notably that of establishing boundedness estimates such as for linear (or “linear-like”) operators from one Lebesgue space to another . (Interpolation can also be performed for many other normed vector spaces than the Lebesgue spaces, but we will just focus on Lebesgue spaces in these notes to focus the discussion.) Using interpolation, it is possible to reduce the task of proving such estimates to that of proving various “endpoint” versions of these estimates. In some cases, each endpoint only faces a portion of the difficulty that the interpolated estimate did, and so by using interpolation one has split the task of proving the original estimate into two or more simpler subtasks. In other cases, one of the endpoint estimates is very easy, and the other one is significantly more difficult than the original estimate; thus interpolation does not really simplify the task of proving estimates in this case, but at least clarifies the relative difficulty between various estimates in a given family.
As is the case with many other tools in analysis, interpolation is not captured by a single “interpolation theorem”; instead, there are a family of such theorems, which can be broadly divided into two major categories, reflecting the two basic methods that underlie the principle of interpolation. The real interpolation method is based on a divide and conquer strategy: to understand how to obtain control on some expression such as for some operator and some function , one would divide into two or more components, e.g. into components where is large and where is small, or where is oscillating with high frequency or only varying with low frequency. Each component would be estimated using a carefully chosen combination of the extreme estimates available; optimising over these choices and summing up (using whatever linearity-type properties on are available), one would hope to get a good estimate on the original expression. The strengths of the real interpolation method are that the linearity hypotheses on can be relaxed to weaker hypotheses, such as sublinearity or quasilinearity; also, the endpoint estimates are allowed to be of a weaker “type” than the interpolated estimates. On the other hand, the real interpolation often concedes a multiplicative constant in the final estimates obtained, and one is usually obligated to keep the operator fixed throughout the interpolation process. The proofs of real interpolation theorems are also a little bit messy, though in many cases one can simply invoke a standard instance of such theorems (e.g. the Marcinkiewicz interpolation theorem) as a black box in applications.
The complex interpolation method instead proceeds by exploiting the powerful tools of complex analysis, in particular the maximum modulus principle and its relatives (such as the Phragmén-Lindelöf principle). The idea is to rewrite the estimate to be proven (e.g. ) in such a way that it can be embedded into a family of such estimates which depend holomorphically on a complex parameter in some domain (e.g. the strip . One then exploits things like the maximum modulus principle to bound an estimate corresponding to an interior point of this domain by the estimates on the boundary of this domain. The strengths of the complex interpolation method are that it typically gives cleaner constants than the real interpolation method, and also allows the underlying operator to vary holomorphically with respect to the parameter , which can significantly increase the flexibility of the interpolation technique. The proofs of these methods are also very short (if one takes the maximum modulus principle and its relatives as a black box), which make the method particularly amenable for generalisation to more intricate settings (e.g. multilinear operators, mixed Lebesgue norms, etc.). On the other hand, the somewhat rigid requirement of holomorphicity makes it much more difficult to apply this method to non-linear operators, such as sublinear or quasilinear operators; also, the interpolated estimate tends to be of the same “type” as the extreme ones, so that one does not enjoy the upgrading of weak type estimates to strong type estimates that the real interpolation method typically produces. Also, the complex method runs into some minor technical problems when target space ceases to be a Banach space (i.e. when ) as this makes it more difficult to exploit duality.
Despite these differences, the real and complex methods tend to give broadly similar results in practice, especially if one is willing to ignore constant losses in the estimates or epsilon losses in the exponents.
The theory of both real and complex interpolation can be studied abstractly, in general normed or quasi-normed spaces; see e.g. this book for a detailed treatment. However in these notes we shall focus exclusively on interpolation for Lebesgue spaces (and their cousins, such as the weak Lebesgue spaces and the Lorentz spaces ).
— 1. Interpolation of scalars —
As discussed in the introduction, most of the interesting applications of interpolation occur when the technique is applied to operators . However, in order to gain some intuition as to why interpolation works in the first place, let us first consider the significantly simpler (though rather trivial) case of interpolation in the case of scalars or functions.
One can view and as the unique log-linear functions of (i.e. , are (affine-)linear functions of ) which equal their boundary values and respectively as .
Example 1 If and for some and , then the log-linear interpolant is given by , where is the quantity such that .
The deduction of (3) from (1), (2) is utterly trivial, but there are still some useful lessons to be drawn from it. For instance, let us take for simplicity, so we are interpolating two upper bounds , on the same quantity to give a new bound . But actually we have a refinement available to this bound, namely
for any sufficiently small (indeed one can take any less than or equal to ). Indeed one sees this simply by applying (3) with with and and taking minima. Thus we see that (3) is only sharp when the two original bounds are comparable; if instead we have for some integer , then (6) tells us that we can improve (3) by an exponentially decaying factor of . The geometric series formula tells us that such factors are absolutely summable, and so in practice it is often a useful heuristic to pretend that the cases dominate so strongly that the other cases can be viewed as negligible by comparison.
Also, one can trivially extend the deduction of (3) from (1), (2) as follows: if is a function from to which is log-convex (thus is a convex function of , and (1), (2) hold for some , then (3) holds for all intermediate also, where is of course defined by (5). Thus one can interpolate upper bounds on log-convex functions. However, one certainly cannot interpolate lower bounds: lower bounds on a log-convex function at and yield no information about the value of, say, . Similarly, one cannot extrapolate upper bounds on log-convex functions: an upper bound on, say, and does not give any information about . (However, an upper bound on coupled with a lower bound on gives a lower bound on ; this is the contrapositive of an interpolation statement.)
Exercise 2 Show that the sum , product , or pointwise maximum of two log-convex functions is log-convex.
Remark 3 Every non-negative log-convex function is convex, thus in particular for all (note that this generalises the arithmetic mean-geometric mean inequality). Of course, the converse statement is not true.
Now we turn to the complex version of the interpolation of log-convex functions, a result known as Lindelöf’s theorem:
for all and some constants . Suppose also that and for all . Then we have for all and , where is of course defined by (5).
Remark 5 The hypothesis (7) is a qualitative hypothesis rather than a quantitative one, since the exact values of do not show up in the conclusion. It is quite a mild condition; any function of exponential growth in , or even with such super-exponential growth as or , will obey (7). The principle however fails without this hypothesis, as one can see for instance by considering the holomorphic function .
Proof: Observe that the function is holomorphic and non-zero on , and has magnitude exactly on the line for each . Thus, by dividing by this function (which worsens the qualitative bound (7) slightly) we may reduce to the case when for all .
Suppose we temporarily assume that as . Then by the maximum modulus principle (applied to a sufficiently large rectangular portion of the strip), it must then attain a maximum on one of the two sides of the strip. But on these two sides, and so on the interior as well.
To remove the assumption that goes to zero at infinity, we use the trick of giving ourselves an epsilon of room. Namely, we multiply by the holomorphic function for some . A little complex arithmetic shows that the function goes to zero at infinity in (the factor decays fast enough to damp out the growth of as , while the damps out the growth as ), and is bounded in magnitude by on both sides of the strip . Applying the previous case to this function, then taking limits as , we obtain the claim.
Exercise 6 With the notation and hypotheses of Theorem 4, show that the function is log-convex on .
Exercise 7 (Hadamard three-circles theorem) Let be a holomorphic function on an annulus . Show that the function is log-convex on .
Exercise 8 (Phragmén-Lindelöf principle) Let be as in Theorem 4, but suppose that we have the bounds and for all and some exponents and a constant . Show that one has for all and some constant (which is allowed to depend on the constants in (7)). (Hint: it is convenient to work first in a half-strip such as for some large . Then multiply by something like for some suitable branch of the logarithm and apply a variant of Theorem 4 for the half-strip. A more refined estimate in this regard is due to Rademacher.) This particular version of the principle gives the convexity bound for Dirichlet series such as the Riemann zeta function. Bounds which exploit the deeper properties of these functions to improve upon the convexity bound are known as subconvexity bounds and are of major importance in analytic number theory, which is of course well outside the scope of this course.
— 2. Interpolation of functions —
We now turn to the interpolation in function spaces, focusing particularly on the Lebesgue spaces and the weak Lebesgue spaces . Here, is a fixed measure space. It will not matter much whether we deal with real or complex spaces; for sake of concretness we work with complex spaces. Then for , recall (see 245B Notes 3) that is the space of all functions whose norm
is finite, modulo almost everywhere equivalence. The space is defined similarly, but where is the essential supremum of on .
A simple test case in which to understand the norms better is that of a step function , where is a non-negative number and a set of finite measure. Then one has for . Observe that this is a log-convex function of . This is a general phenomenon:
for all , where the exponent is defined by .
In particular, we see that the function is log-convex whenever the right-hand side is finite (and is in fact log-convex for all , if one extends the definition of log-convexity to functions that can take the value ). In other words, we can interpolate any two bounds and to obtain for all .
Let us give several proofs of this lemma. We will focus on the case ; the endpoint case can be proven directly, or by modifying the arguments below, or by using an appropriate limiting argument, and we leave the details to the reader.
The first proof is to use Hölder’s inequality
when is finite (with some minor modifications in the case ).
Another (closely related) proof proceeds by using the log-convexity inequality
for all , where is the quantity such that . If one integrates this inequality in , one already obtains the claim in the normalised case when . To obtain the general case, one can multiply the function and the measure by appropriately chosen constants to obtain the above normalisation; we leave the details as an exercise to the reader. (The case when or vanishes is of course easy to handle separately.)
A third approach is more in the spirit of the real interpolation method, avoiding the use of convexity arguments. As in the second proof, we can reduce to the normalised case . We then split , where is the indicator function to the set , and similarly for . Observe that
and so by the quasi-triangle inequality (or triangle inequality, when )
This is off by a constant factor by what we want. But one can eliminate this constant by using the tensor power trick. Indeed, if one replaces with a Cartesian power (with the product -algebra and product measure ), and replace by the tensor power , we see from many applications of the Fubini-Tonelli theorem that
for all . In particular, obeys the same normalisation hypotheses as , and thus by applying the previous inequality to , we obtain
for every , where it is key to note that the constant on the right is independent of . Taking roots and then sending , we obtain the claim.
Finally, we give a fourth proof in the spirit of the complex interpolation method. By replacing by we may assume is non-negative. By expressing non-negative measurable functions as the monotone limit of simple functions and using the monotone convergence theorem, we may assume that is a simple function, which is then necessarily of finite measure support from the finiteness hypotheses. Now consider the function . Expanding out in terms of step functions we see that this is an analytic function of which grows at most exponentially in ; also, by the triangle inequality this function has magnitude at most when and magnitude when . Applying Theorem 4 and specialising to the value of for which we obtain the claim.
Exercise 10 If , show that equality holds in Lemma 9 if and only if is a step function.
Now we consider variants of interpolation in which the “strong” spaces are replaced by their “weak” counterparts . Given a measurable function , we define the distribution function by the formula
This distribution function is closely connected to the norms. Indeed, from the calculus identity
for all , thus the norms are essentially moments of the distribution function. The norm is of course related to the distribution function by the formula
for any measurable and , where we use to denote a pair of inequalities of the form for some constants depending only on . (Hint: is non-increasing in .) Thus we can relate the norms of to the dyadic values of the distribution function; indeed, for any , is comparable (up to constant factors depending on ) to the norm of the sequence .
Another relationship between the norms and the distribution function is given by observing that
for any , leading to Chebyshev’s inequality
(The version of this inequality is also known as Markov’s inequality. In probability theory, Chebyshev’s inequality is often specialised to the case , and with replaced by a normalised function . Note that, as with many other Cyrillic names, there are also a large number of alternative spellings of Chebyshev in the Roman alphabet.)
Chebyshev’s inequality motivates one to define the weak norm of a measurable function for by the formula
thus Chebyshev’s inequality can be expressed succinctly as
It is also natural to adopt the convention that . If are two functions, we have the inclusion
this easily leads to the quasi-triangle inequality
where we use as shorthand for the inequality for some constant depending only on (it can be a different constant at each use of the notation). [Note: in analytic number theory, it is more customary to use instead of , following Vinogradov. However, in analysis is sometimes used instead to denote “much smaller than”, e.g. denotes the assertion for some sufficiently small constant .]
Let be the space of all which have finite , modulo almost everywhere equivalence; this space is also known as weak . The quasi-triangle inequality soon implies that is a quasi-normed vector space with the quasi-norm, and Chebyshev’s inequality asserts that contains as a subspace (though the norm is not a restriction of the norm).
Example 12 If with the usual measure, and , then the function is in weak , but not strong . It is also not in strong or weak for any other . But the “local” component of is in strong and weak for all , and the “global” component of is in strong and weak for all .
Exercise 13 For any and , define the (dyadic) Lorentz norm to be norm of the sequence , and define the Lorentz space be the space of functions with finite, modulo almost everywhere equivalence. Show that is a quasi-normed space, which is equivalent to when and to when . Lorentz spaces arise naturally in more refined applications of the real interpolation method, and are useful in certain “endpoint” estimates that fail for Lebesgue spaces, but which can be rescued by using Lorentz spaces instead. However, we will not pursue these applications in detail here.
Exercise 14 Let be a finite set with counting measure, and let be a function. For any , show that
(Hint: to prove the second inequality, normalise , and then manually dispose of the regions of where is too large or too small.) Thus, in some sense, weak and strong are equivalent “up to logarithmic factors”.
for all . Indeed, from the hypotheses we have
for all .
As remarked in the previous section, we can improve upon (11); indeed, if we define to be the unique value of where and are equal, then we have
for some constant . Note that one cannot use the tensor power trick this time to eliminate the constant as the weak norms do not behave well with respect to tensor product. Indeed, the constant must diverge to infinity in the limit if , otherwise it would imply that the norm is controlled by the norm, which is false by Example 12; similarly one must have a divergence as if .
Exercise 15 Let and . Refine the inclusions in (8) to
Define the strong type diagram of a function to be the set of all for which lies in strong , and the weak type diagram to be the set of all for which lies in weak . Then both the strong and weak type diagrams are connected subsets of , and the strong type diagram is contained in the weak type diagram, and contains in turn the interior of the weak type diagram. By experimenting with linear combinations of the examples in Example 12 we see that this is basically everything one can say about the strong and weak type diagrams, without further information on or .
Exercise 16 Let be a measurable function which is finite almost everywhere. Show that there exists a unique non-increasing left-continuous function such that for all , and in particular for all , and . (Hint: first look for the formula that describes for some in terms of .) The function is known as the non-increasing rearrangement of , and the spaces and are examples of rearrangement-invariant spaces. There are a class of useful rearrangement inequalities that relate to its rearrangements, and which can be used to clarify the structure of rearrangement-invariant spaces, but we will not pursue this topic here.
- lies in , thus for some finite .
- There exists a constant such that for all sets of finite measure.
Furthermore show that the best constants in the above statements are equivalent up to multiplicative constants depending on , thus . Conclude that the modified weak norm , where ranges over all sets of positive finite measure, is a genuine norm on which is equivalent to the quasinorm.
Exercise 18 Let be an integer. Find a probability space and functions with for such that for some absolute constant . (Hint: exploit the logarithmic divergence of the harmonic series .) Conclude that there exists a probability space such that the quasi-norm is not equivalent to an actual norm.
Exercise 19 Let be a -finite measure space, let , and be a measurable function. Show that the following are equivalent:
- lies in .
- There exists a constant such that for every set of finite measure, there exists a subset with such that .
Exercise 20 Let be a measure space of finite measure, and be a measurable function. Show that the following two statements are equivalent:
- There exists a constant such that for all .
- There exists a constant such that .
— 3. Interpolation of operators —
We turn at last to the central topic of these notes, which is interpolation of operators between functions on two fixed measure spaces and . To avoid some (very minor) technicalities we will make the mild assumption throughout that and are both -finite, although much of the theory here extends to the non--finite setting.
A typical situation is that of a linear operator which maps one space to another , and also maps to for some exponents ; thus (by linearity) will map the larger vector space to , and one has some estimates of the form
for all respectively, and some . We would like to then interpolate to say something about how maps to .
The complex interpolation method gives a satisfactory result as long as the exponents allow one to use duality methods, a result known as the Riesz-Thorin theorem:
for all and , where , , and .
Remark 22 When is a point, this theorem essentially collapses to Lemma 9 (and when is a point, this is a dual formulation of that lemma); and when and are both points; this collapses to interpolation of scalars.
Proof: If then the claim follows from Lemma 9, so we may assume , which in particular forces to be finite. By symmetry we can take . By multiplying the measures and (or the operator ) by various constants, we can normalise (the case when or is trivial). Thus we have also.
By Hölder’s inequality, the bound (13) implies that
for all and .
for all , that are simple functions with finite measure support. To see this, we first normalise . Observe that we can write , for some functions of magnitude at most . If we then introduce the quantity
(with the conventions that in the endpoint case ) we see that is a holomorphic function of of at most exponential growth which equals when . When instead , an application of (15) shows that ; a similar claim obtains when using (16). The claim now follows from Theorem 4.
The estimate (17) has currently been established for simple functions with finite measure support. But one can extend the claim to any (keeping simple with finite measure support) by decomposing into a bounded function and a function of finite measure support, approximating the former in by simple functions of finite measure support, and approximating the latter in by simple functions of finite measure support, and taking limits using (15), (16) to justify the passage to the limit. One can then also allow arbitrary by using the monotone convergence theorem. The claim now follows from the duality between and .
Suppose one has a linear operator that maps simple functions of finite measure support on to measurable functions on (modulo almost everywhere equivalence). We say that such an operator is of strong type if it can be extended in a continuous fashion to an operator on to an operator on ; this is equivalent to having an estimate of the form for all simple functions of finite measure support. (The extension is unique if is finite or if has finite measure, due to the density of simple functions of finite measure support in those cases. Annoyingly, uniqueness fails for of an infinite measure space, though this turns out not to cause much difficulty in practice, as the conclusions of interpolation methods are usually for finite exponents .) Define the strong type diagram to be the set of all such that is of strong type . The Riesz-Thorin theorem tells us that if is of strong type and with and , then is also of strong type for all ; thus the strong type diagram contains the closed line segment connecting with . Thus the strong type diagram of is convex in at least. (As we shall see later, it is in fact convex in all of .) Furthermore, on the intersection of the strong type diagram with , the operator norm is a log-convex function of .
Exercise 23 If with the usual measure, show that the strong type diagram of the identity operator is the triangle . If instead with the usual counting measure, show that the strong type diagram of the identity operator is the triangle . What is the strong type diagram of the identity when with the usual measure?
Exercise 24 Let (resp. ) be a linear operator from simple functions of finite measure support on (resp. ) to measurable functions on (resp. ) modulo a.e. equivalence that are absolutely integrable on finite measure sets. We say are formally adjoint if we have for all simple functions of finite measure support on respectively. If , show that is of strong type if and only if is of strong type . Thus, taking formal adjoints reflects the strong type diagram around the line of duality , at least inside the Banach space region .
Remark 25 There is a powerful extension of the Riesz-Thorin theorem known as the Stein interpolation theorem, in which the single operator is replaced by a family of operators for that vary holomorphically in in the sense that is a holomorphic function of for any sets of finite measure. Roughly speaking, the Stein interpolation theorem asserts that if is of strong type for with a bound growing at most exponentially in , and itself grows at most exponentially in in some sense, then will be of strong type . A precise statement of the theorem and some applications can be found in Stein’s book on harmonic analysis.
Now we turn to the real interpolation method. Instead of linear operators, it is now convenient to consider sublinear operators mapping simple functions of finite measure support in to -valued measurable functions on (modulo almost everywhere equivalence, as usual), obeying the homogeneity relationship
and the pointwise bounds
for all , and all simple functions of finite measure support.
Every linear operator is sublinear; also, the absolute value of a linear (or sublinear) operator is also sublinear. More generally, any maximal operator of the form , where is a family of sub-linear operators, is also a non-negative sublinear operator; note that one can also replace the supremum here by any other norm in , e.g. one could take an norm for any . (After and , a particularly common case is when , in which case is known as a square function.)
The basic theory of sublinear operators is similar to that of linear operators in some respects. For instance, continuity is still equivalent to boundedness:
Exercise 26 Let be a sublinear operator, and let . Assume that either is finite, or has finite measure. Then the following are equivalent:
- can be extended to a continuous operator from to .
- There exists a constant such that for all simple functions of finite measure support.
- can be extended to a operator from to such that for all and some .
Show that the extension mentioned above is unique. Finally, show that the same equivalences hold if is replaced by throughout.
We say that is of strong type if any of the above equivalent statements (for ) hold, and of weak type if any of the above equivalent statements (for ) hold. We say that a linear operator is of strong or weak type if its non-negative counterpart is; note that this is compatible with our previous definition of strong type for such operators. Also, Chebyshev’s inequality tells us that strong type implies weak type .
We now give the real interpolation counterpart of the Riesz-Thorin theorem, namely the Marcinkeiwicz interpolation theorem:
Theorem 27 (Marcinkiewicz interpolation theorem) Let and be such that , and for . Let be a sublinear operator which is of weak type and of weak type . Then is of strong type .
Remark 28 Of course, the same claim applies to linear operators by setting . One can also extend the argument to quasilinear operators, in which the pointwise bound is replaced by for some constant , but this generalisation only appears occasionally in applications. The conditions can be replaced by the variant condition (see Exercise 31, Exercise 33), but cannot be eliminated entirely: see Exercise 32. The precise hypotheses required on are rather technical and I recommend that they be ignored on a first reading.
Proof: For notational reasons it is convenient to take finite; however the arguments below can be modified without much difficulty to deal with the infinite case (or one can use a suitable limiting argument); we leave this to the interested reader.
for all simple functions of finite measure support, and all . Let us write to denote for some constant depending on the indicated parameters. By (9), it will suffice to show that
By homogeneity we can normalise .
see Exercise 11. The hypothesis similarly implies that
When , the claim follows from direct substitution of (18), (19) (see also the discussion in the previous section about interpolating strong bounds from weak ones), so let us assume ; by symmetry we may take , and thus . In this case we cannot directly apply (18), (19) because we only control in , not or . To get around this, we use the basic real interpolation trick of decomposing into pieces. There are two basic choices for what decomposition to pick. On one hand, one could adopt a “minimalistic” approach and just decompose into two pieces
where and , and the threshold is a parameter (depending on ) to be optimised later. Or we could adopt a “maximalistic” approach and perform the dyadic decomposition
where . (Note that only finitely many of the are non-zero, as we are assuming to be a simple function.) We will adopt the latter approach, in order to illustrate the dyadic decomposition method; the former approach also works, but we leave it as an exercise to the interested reader.
From sublinearity we have the pointwise estimate
which implies that
whenever are positive constants such that , but for which we are otherwise at liberty to choose. We will set aside the problem of deciding what the optimal choice of is for now, and continue with the proof.
From construction of we can bound
and similarly for , and thus we have
for . To prove (20), it thus suffices to show that
It is convenient to introduce the quantities appearing in (21), thus
and our task is to show that
for all .
We can simplify this expression a bit by collecting terms and making some substitutions. The points are collinear, and we can capture this by writing
for some and some . We can then simplify the left-hand side of (22) to
Note that is positive and is negative. If we then pick to be a sufficiently small multiple of where (say), we obtain the claim by summing geometric series.
Remark 29 A closer inspection of the proof (or a rescaling argument to reduce to the normalised case , as in preceding sections) reveals that one establishes the estimate
for all simple functions of finite measure support (or for all , if one works with the continuous extension of to such functions), and some constant . Thus the conclusion here is weaker by a multiplicative constant from that in the Riesz-Thorin theorem, but the hypotheses are weaker too (weak-type instead of strong-type). Indeed, we see that the constant must blow up as or .
The power of the Marcinkiewicz interpolation theorem, as compared to the Riesz-Thorin theorem, is that it allows one to weaken the hypotheses on from strong type to weak type. Actually, it can be weakened further. We say that a non-negative sublinear operator is restricted weak-type for some if there is a constant such that
for all sets of finite measure and all simple functions with . Clearly restricted weak-type is implied by weak-type , and thus by strong-type . (One can also define the notion of restricted strong-type by replacing with ; this is between strong-type and restricted weak-type , but is incomparable to weak-type .)
Exercise 30 Show that the Marcinkiewicz interpolation theorem continues to hold if the weak-type hypotheses are replaced by restricted weak-type hypothesis. (Hint: where were the weak-type hypotheses used in the proof?)
We thus see that the strong-type diagram of contains the interior of the restricted weak-type or weak-type diagrams of , at least in the triangular region .
Exercise 31 Suppose that is a sublinear operator of restricted weak-type and for some . Show that is of restricted weak-type for any , or in other words the restricted type diagram is convex in . (This is an easy result requiring only interpolation of scalars.) Conclude that the hypotheses in the Marcinkiewicz interpolation theorem can be replaced by the variant .
Exercise 32 For any , let be the natural numbers with the weighted counting measure , thus each point has mass . Show that if , then the identity operator from to is of weak-type but not strong-type when and . Conclude that the hypotheses cannot be dropped entirely.
for all simple functions of finite measure support, where the Lorentz norms were defined in Exercise 13. (Hint: repeat the proof of the Marcinkiewicz interpolation theorem, but partition the sum into regions of the form for integer . Obtain a bound for each summand which decreases geometrically as .) Conclude that the hypotheses in the Marcinkiewicz interpolation theorem can be replaced by . This Lorentz space version of the interpolation theorem is in some sense the “right” version of the theorem, but the Lorentz spaces are slightly more technical to deal with than the Lebesgue spaces, and the Lebesgue space version of Marcinkiewicz interpolation is largely sufficient for most applications.
Exercise 34 For , let be -finite measure spaces, and let be a linear operator from simple functions of finite measure support on to measurable functions on (modulo almost everywhere equivalence, as always). Let , be the product spaces (with product -algebra and product measure). Show that there exists a unique (modulo a.e. equivalence) linear operator defined on linear combinations of indicator functions of product sets of sets , of finite measure, such that
for a.e. ; we refer to as the tensor product of and and write . Show that if are of strong-type for some with operator norms respectively, then can be extended to a bounded linear operator on to with operator norm exactly equal to , thus
(Hint: for the lower bound, show that for all simple functions . For the upper bound, express as the composition of two other operators and for some identity operators , and establish operator norm bounds on these two operators separately.) Use this and the tensor power trick to deduce the Riesz-Thorin theorem (in the special case when for , and ) from the Marcinkiewicz interpolation theorem. Thus one can (with some effort) avoid the use of complex variable methods to prove the Riesz-Thorin theorem, at least in some cases.
Exercise 35 (Hölder’s inequality for Lorentz spaces) Let and for some . Show that , where and , with the estimate
for some constant . (This estimate is due to O’Neil.)
Remark 36 Just as interpolation of functions can be clarified by using step functions as a test case, it is instructive to use rank one operators such as
where are finite measure sets, as test cases for the real and complex interpolation methods. (After understanding the rank one case, I then recommend looking at the rank two case, e.g. , where could be very different in size from .)
— 4. Some examples of interpolation —
Now we apply the interpolation theorems to some classes of operators. An important such class is given by the integral operators
from functions to functions , where is a fixed measurable function, known as the kernel of the integral operator . Of course, this integral is not necessarily convergent, so we will also need to study the sublinear analogue
which is well-defined (though it may be infinite).
The following useful lemma gives us strong-type bounds on and hence , assuming certain type bounds on the rows and columns of .
Lemma 37 (Schur’s test) Let be a measurable function obeying the bounds
for almost every , and
for almost every , where and . Then for every , and are of strong-type , with well-defined for all and almost every , and furthermore
Here we adopt the convention that and , thus and .
Proof: The hypothesis , combined with Minkowski’s integral inequality, shows us that
for all ; in particular, for such , is well-defined almost everywhere, and
Similarly, Hölder’s inequality tells us that for , is well-defined everywhere, and
Applying the Riesz-Thorin theorem we conclude that
for all simple functions with finite measure support; replacing with we also see that
for all simple functions with finite measure support, and thus (by monotone convergence) for all . The claim then follows.
for all vectors and all . Note the extreme cases , can be seen directly; the remaining cases then follow from interpolation.
A useful special case arises when is an -sparse matrix, which means that at most entries in any row or column are non-zero (e.g. permutation matrices are -sparse). We then conclude that the operator norm of is at most .
Exercise 39 Establish Schur’s test by more direct means, taking advantage of the duality relationship
A useful corollary of Schur’s test is Young’s convolution inequality for the convolution of two functions , , defined as
provided of course that the integrand is absolutely convergent.
Exercise 40 (Young’s inequality) Let be such that . Show that if and , then is well-defined almost everywhere and lies in , and furthermore that
(Hint: Apply Schur’s test to the kernel .)
Remark 41 There is nothing special about here; one could in fact use any locally compact group with a bi-invariant Haar measure. On the other hand, if one specialises to , then it is possible to improve Young’s inequality slightly, to
where , a result of Beckner; the constant here is best possible, as can be seen by testing the inequality in the case when are Gaussians.
Exercise 42 Let , and let , . Young’s inequality tells us that . Refine this further by showing that , i.e. is continuous and goes to zero at infinity. (Hint: first show this when , then use a limiting argument.)
We now give a variant of Schur’s test that allows for weak estimates.
Lemma 43 (Weak-type Schur’s test) Let be a measurable function obeying the bounds
for almost every , and
for almost every , where and (note the endpoint exponents are now excluded). Then for every , and are of strong-type , with well-defined for all and almost every , and furthermore
Here we again adopt the convention that and .
Proof: From Exercise 17 we see that
for any measurable , where we use to denote for some depending on the indicated parameters. By the Fubini-Tonelli theorem, we conclude that
for any ; by Exercise 17 again we conclude that
thus is of weak-type . In a similar vein, from yet another application of Exercise 17 we see that
whenever and has finite measure; thus is of restricted type . Applying Exercise 30 we conclude that is of strong type (with operator norm ), and the claim follows.
This leads to a weak-type version of Young’s inequality:
Exercise 44 (Weak-type Young’s inequality) Let be such that . Show that if and , then is well-defined almost everywhere and lies in , and furthermore that
for some constant .
Exercise 45 Refine the previous exercise by replacing with the Lorentz space throughout.
Recall that the function will lie in for . We conclude
is well-defined almost everywhere and lies in , and furthermore that
for some constant .
This inequality is of importance in the theory of Sobolev spaces, which we will discuss in a subsequent lecture.
Exercise 47 Show that Corollary 46 can fail at the endpoints , , or .
Update, Apr 6: another exercise added; note renumbering.
Update, Apr 8: some formatting errors fixed.
Update, Sep 14: definition of sublinearity fixed.