An A_p – A_\infty Conjecture

The weighted theory, started by Benjamim Muckenhoupt, which came of age with the result of Hunt-Muckenhoupt-Wheeden concerns { A_p} weights. These are defined by

Definition 1 Let { w} be a weight with density also written as { w}. Assume { w>0} a.\thinspace e.\thinspace, and { 1<p< \infty }. We define { \sigma ^{1-p'}}, which is defined a.\thinspace e.\thinspace, and set

\displaystyle   \| w\|_{A_p} := \sup _{Q} \frac {w (Q)} {\lvert Q\rvert } \Bigl[ \frac {\sigma (Q)} {\lvert Q\rvert } \Bigr] ^{p-1} \,. \ \ \ \ \ (1)

For the two endpoints { p=1}, and { p= \infty }, we set \| w\|_{A_1}& := \Bigl\| \frac {Mw} {w} \Bigr\|_{\infty }
\| w\|_{A_ \infty }& := \sup _{Q} w (Q) ^{-1} \int _{Q} M (w Q) \; dx \,.

The definition is easiest to understand when { p=2}. The condition { w >0} a.\thinspace e.\thinspace implies that { \sigma = w ^{-1} } is defined almost everywhere, and moreover { w \cdot \sigma \equiv 1}. The condition of { w\in A_2} implies that this equality holds in an average sense, uniformly of locations and scales. The definition of { A_1} is the limiting case of (1). The endpoint { p= \infty } is a little more subtle.

Theorem 2 (Muckenhoupt) Let { w >0 } a.\thinspace e., and let { 1< p < \infty }. Then the following three conditions are equivalent.

  • The Maximal Function is bounded on { L ^{p} (w)} into itself;
  • The Maximal Function is bounded on { L ^{p} (w)} into { L ^{p, \infty } (w)};
  • { w\in A_p}.

Of course the difficult part is to show the sufficiency of { w \in A_p} for the boundedness of the Maximal function. For the Hilbert transform, however, both directions require arguement.

Theorem 3 (Hunt-Muckenhoupt-Wheeden) In dimension one, let { w>0} a.\thinspace e. Then, the three equivalences above hold, with the Maximal function replaced by the Hilbert transform.

Finer information about the norm of { M} or { H} is useful in applying these inequalities, and illuminates the relationship between the operator and the weights. This theme goes back to Buckley

Theorem 4 (Buckley) Let { 1< p < \infty }, and let { w \in A_p}, and { \sigma = w ^{-p'+1}} the dual measure. We have

\displaystyle  \| M _{\sigma } f \|_{ L ^{p, \infty } (w)} \lesssim \| w\|_{ A_p} ^{1/p} \| f\|_{ L ^{p} (\sigma )} \,,

\displaystyle   \| M _{\sigma } f \|_{ L ^{p } (w)} \lesssim \| w\|_{ A_p} ^{1/(p-1)} \| f\|_{ L ^{p} (\sigma )} \,,

We have the following refinement of the Buckley’s estimate, due to Hytönen-Perez.

Theorem 5 Given two weights { w , \sigma } we have

\displaystyle   \| M _{\sigma } f \|_{ L ^{p } (w)} \lesssim \| w, \sigma \| _{A_p} \| \sigma \|_{A _{\infty }} ^{1/p} \| f\|_{ L ^{p} (\sigma )}

Establishing the same results for singular integrals. There was a breakthrough by Stefanie Petermichl, establishing the sharp bounds for the Hilbert transform. Much later, Lacey-Petermichl-Reguera gave a more conceptual argument, greatly extending the range of Petermichl’s theorem. This proof, with two more important new ingredients, was used by Hytönen to prove

Theorem 6 For { T } an { L ^{2} (\mathbb R ^{d})} bounded Calderón-Zygmund Operator and {1<p<\infty},

\displaystyle   \| T f \| _{ L ^{p,\infty } (w)} \le C_{T,p} \| w \|_{ A _{p}}^{ \max \{1, (p-1) ^{-1} \}} \| f \| _{L ^{p} (w)},

Also see Hytönen-Perez-Treil-Volberg.

This type of result has now been extended to maximal truncations, in a seven author paper H-L-M-O-R-S-(U-T). (Hyt\”onen, Lacey, Martikainen, Orponen, Reguera, Sawyer, and Uriate-Tuero.
There is a seperate post about the details of this paper, and how it came to be written. But, the post is not yet written.)

This is the strongest inequalities currently known, depending only upon the { A_p} characteristic. Below, { T _{\natural} } stands for the maximal truncations of { T}.

Theorem 7 For { T } an { L ^{2} (\mathbb R ^{d})} bounded Calderón-Zygmund Operator,

\displaystyle   \| T _{\natural } f \| _{ L ^{p,\infty } (w)} \le C_T \| w \| _{ A _{p}} \| f \| _{L ^{p} (w)} \, \,,\qquad 1 < p < 2

\displaystyle     \| T _{\natural} f \| _{ L ^{p } (w)} \le C_T \| w \| _{ A _{p}} ^{ \max \{1, (p-1) ^{-1}\} } \| f \| _{L ^{p} (w)} \, \,, \qquad 1 < p < \infty \,.

The paper of Lacey-Petermichl-Reguera nicely divided the proof of these sorts of estimates into an { A_p} component, and an { A _{\infty }} component. And, the paper Lerner quantified this point of view. The paper of Hytönen-Perez identifies this as an important theme, and they prove

Theorem 8 For { T } an { L ^{2} (\mathbb R ^{d})} bounded Calderón-Zygmund Operator,

\displaystyle   \| T  f \| _{ L ^{2 } (w)} \le C_T \| w \| _{ A _{2}} ^{1/2 } \max \{ \| w \| _{ A _{\infty }} , \| \sigma \| _{ A _{\infty }} \} ^{1/2} \| f \| _{L ^{p} (w)} \,.

The following estimate is proved in H-L-M-O-R-S-(U-T).

Theorem 9 For { T } an { L ^{2} (\mathbb R ^{d})} bounded Calderón-Zygmund Operator and {1<p<\infty},

\displaystyle   \| T _{\natural } f \| _{ L ^{p,\infty } (w)} \le C_{T,p} \| w \|_{ A _{p}}^{1/p} \| w \|_{ A _{\infty}}^{1/p'}  \| f \| _{L ^{p} (w)},

A formalism from Eric Saywer’s fractional integral estimates then makes it natural to pose the conjecture below. [Update: This formalism is that the strong type norm is the maximum of two
weak-type norms.]

Conjecture 10 For { T } an { L ^{2} (\mathbb R ^{d})} bounded Calderón-Zygmund Operator and {1<p<\infty},

\displaystyle  \| T _{\natural } f \| _{ L ^{p} (w)} \le C_{T,p} \| w \|_{ A _{p}}^{1/p} \max \bigl\{ \| w \|_{ A _{\infty}}^{1/p'}, \| w ^{- p'+1} \|_{A _{\infty } } ^{1/p} \bigr\} \| f \| _{L ^{p} (w)}.\,.

Tuomas Hytönen pointed out the nice interpretation of this conjecture to me. It is familiar that one typically has { \| w \|_{ A _{\infty}}^{1/p}} as a a lower bound on the norm of { T}, and so the term above identifies the `{A _{\infty } } contribution' to the norm of { T}.

Using the Lerner Median inequality, and the main theorem of Lacey-Sawyer-Uriate-Tuero, I can verify this for Haar shift operators, with an exponential dependence on complexity. And I will present this proof at the Paesky Spring School.

The paper H-L-M-O-R-S-(U-T) gives an estimate for { T _{\natural}}, in strong type norm, which verifies the conjecture in the special case that

\displaystyle    \| w\|_{A _{\infty }} \gtrsim \| w\|_{A_p} ^{1/ p ^2 }, \qquad p>2\,.

If there were some further refinement of the proof of the two-weight result
H-L-M-O-R-S-(U-T), Theorem 4.7, then maybe the Conjecture would follow. Namely the obstacle is the use of the ‘non-standard’ testing condition in (4.8) of this paper.