Published by De Gruyter December 8, 2017

Conical square functions for degenerate elliptic operators

Li Chen , José María Martell and Cruz Prisuelos-Arribas

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2016-0062

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Abstract

The aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let Lw=-w-1⁢div(w⁢A⁢∇), where w∈A2 and A is an n×n bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the L2⁢(w)-Kato square root problem obtaining that Lw is equivalent to the gradient on L2⁢(w). The same authors in collaboration with the second named author of this paper studied the Lp⁢(w)-boundedness of operators that are naturally associated with Lw, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in Lp⁢(v⁢d⁢w) for v∈A∞⁢(w)), and in particular a class of “degeneracy” weights w was found in such a way that the classical L2-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on Lp⁢(w) and on Lp⁢(v⁢d⁢w), with v∈A∞⁢(w), of the conical square functions that one can construct using the heat or Poisson semigroup associated with Lw. As a consequence of our methods, we find a class of degeneracy weights w for which L2-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with Lw.

Keywords: Conical square functions; change of angle formulas; Muckenhoupt weights; degenerate elliptic operators; heat and Poisson semigroups; off-diagonal estimates

MSC 2010: 42B25; 35J15; 35J70; 47G10; 47D06; 42B37; 42B20

Communicated by Kaj Nyström

Funding source: FP7 Ideas: European Research Counci

Award Identifier / Grant number: 615112 HAPDEGMT

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: SEV-2015-0554

Funding statement: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. The authors also acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554).

A Extrapolation on weighted measure spaces

In this section we present some extrapolation results where the underlying measure space is (ℝn,w) with w∈A∞. The statements and proofs are quite similar to the euclidean setting with the Lebesgue measure. As in [12], we write the extrapolation theorem in terms of pairs of functions. To set the stage, consider ℱ a family of pairs (f,g) of non-negative, measurable functions that are not identically zero. Given such a family ℱ, 0<p<∞, and a weight v∈A∞⁢(w), when we write

∫ℝnf⁢(x)p⁢v⁢(x)⁢𝑑w⁢(x)≤Cw,v,p⁢∫ℝng⁢(x)p⁢v⁢(x)⁢𝑑w⁢(x),(f,g)∈ℱ,

we mean that this inequality holds for all pairs (f,g)∈ℱ and that the constant Cw,v,p depends only on p, the A∞⁢(w) constant of v (and the A∞ character of w which is ultimately fixed). Note that in [12] such inequalities appear both in the hypotheses and in the conclusion of the extrapolation results and hold for all pairs (f,g)∈ℱ for which the left-hand sides are finite. Here we do not make such assumptions and, in particular, we do have that the infiniteness of the left-hand side will imply that of the right-hand one. This formulation is more convenient for our purposes and, as pointed out in [27, Section 3.1], it follows from the formulation where the inequalities hold for pairs for which the left-hand sides are finite.

The following result for w=1 can be found in [12, Chapter 2] and [27, Section 3.1]. The proof can be easily obtained by adapting the arguments there replacing everywhere the Lebesgue measure by the measure w and the Hardy–Littlewood maximal function by its “weighted” version ℳw introduced in (2.5). Further details are left to the interested reader.

Theorem A.1.

Let F be a given family of pairs (f,g) of non-negative and not identically zero measurable functions.

Suppose that for some fixed exponent p0, 1≤p0<∞, and every weight v∈Ap0⁢(w),
∫ℝnf⁢(x)p0⁢v⁢(x)⁢𝑑w⁢(x)≤Cw,v,p0⁢∫ℝng⁢(x)p0⁢v⁢(x)⁢𝑑w⁢(x) for all ⁢(f,g)∈ℱ.
Then, for all 1<p<∞, and for all v∈Ap⁢(w),
∫ℝnf⁢(x)p⁢v⁢(x)⁢𝑑w⁢(x)≤Cw,v,p⁢∫ℝng⁢(x)p⁢v⁢(x)⁢𝑑w⁢(x) for all ⁢(f,g)∈ℱ.
Suppose that for some fixed exponent q0, 1≤q0<∞, and every weight v∈R⁢Hq0′⁢(w),
∫ℝnf⁢(x)1q0⁢v⁢(x)⁢𝑑w⁢(x)≤Cw,v,q0⁢∫ℝng⁢(x)1q0⁢v⁢(x)⁢𝑑w⁢(x) for all ⁢(f,g)∈ℱ.
Then, for all 1<q<∞ and for all v∈R⁢Hq′⁢(w),
∫ℝnf⁢(x)1q⁢v⁢(x)⁢𝑑w⁢(x)≤Cw,v,q⁢∫ℝng⁢(x)1q⁢v⁢(x)⁢𝑑w⁢(x) for all ⁢(f,g)∈ℱ.
Suppose that for some fixed exponent r0, 0<r0<∞, and every weight v∈A∞⁢(w),
∫ℝnf⁢(x)r0⁢v⁢(x)⁢𝑑w⁢(x)≤Cw,v,r0⁢∫ℝng⁢(x)r0⁢v⁢(x)⁢𝑑w⁢(x) for all ⁢(f,g)∈ℱ.
Then, for all 0<r<∞ and for all v∈A∞⁢(w),
∫ℝnf⁢(x)r⁢v⁢(x)⁢𝑑w⁢(x)≤Cw,v,r⁢∫ℝng⁢(x)r⁢v⁢(x)⁢𝑑w⁢(x) for all ⁢(f,g)∈ℱ.

The following result is a version of [27, Proposition 3.30] in our current setting.

Proposition A.2.

Let w∈Ar and v∈R⁢Hs′⁢(w) with 1≤r,s<∞. For every 1≤q≤s, 0<α≤1 and t>0, there holds

(A.3)∫ℝn(∫B⁢(x,α⁢t)|h⁢(y,t)|⁢d⁢w⁢(y)w⁢(B⁢(y,α⁢t)))1q⁢v⁢(x)⁢𝑑w⁢(x)≲αn⁢r⁢(1s-1q)⁢∫ℝn(∫B⁢(x,t)|h⁢(y,t)|⁢d⁢w⁢(y)w⁢(B⁢(y,t)))1q⁢v⁢(x)⁢𝑑w⁢(x).

Proof.

We fix t>0, 0<α≤1 and set

Gα⁢(x,t):=∫B⁢(x,α⁢t)|h⁢(y,t)|⁢d⁢w⁢(y)w⁢(B⁢(y,α⁢t)).

For simplicity, G⁢(x,t):=G1⁢(x,t). Then, for any 1≤s0<∞ and v0∈R⁢Hs0′⁢(w), we have

∫ℝnGα⁢(x,t)⁢v0⁢(x)⁢𝑑w⁢(x)=∫ℝn|h⁢(y,t)|⁢v0⁢w⁢(B⁢(y,α⁢t))w⁢(B⁢(y,α⁢t))⁢𝑑w⁢(y)

≲∫ℝn|h⁢(y,t)|⁢v0⁢w⁢(B⁢(y,t))⁢(w⁢(B⁢(y,α⁢t))w⁢(B⁢(y,t)))1s0⁢d⁢w⁢(y)w⁢(B⁢(y,α⁢t))

=∫ℝn∫B⁢(x,t)|h⁢(y,t)|⁢(w⁢(B⁢(y,α⁢t))w⁢(B⁢(y,t)))1s0-1⁢d⁢w⁢(y)w⁢(B⁢(y,t))⁢v0⁢(x)⁢𝑑w⁢(x)

≲αn⁢r⁢(1s0-1)⁢∫ℝn∫B⁢(x,t)|h⁢(y,t)|⁢d⁢w⁢(y)w⁢(B⁢(y,t))⁢v0⁢(x)⁢𝑑w⁢(x)

(A.4)=αn⁢r⁢(1s0-1)⁢∫ℝnG⁢(x,t)⁢v0⁢(x)⁢𝑑w⁢(x).

Note that the two inequalities follow from (2.3) and (2.1), respectively, and the second equality is obtained by using Fubini’s theorem. Let us observe that (A.4) is the desired estimate when q=1.

To prove the case q>1, we next extrapolate from (A.4). Consider ℱ the family of pairs

(f,g)=(Gα⁢(⋅,t)s0,αn⁢r⁢(1-s0)⁢G⁢(⋅,t)s0),

and notice that (A.4) immediately gives that, for every v0∈R⁢Hs0′⁢(w), 1≤s0<∞,

∫ℝnf⁢(x)1s0⁢v0⁢(x)⁢𝑑w⁢(x)≤C⁢∫ℝng⁢(x)1s0⁢v0⁢(x)⁢𝑑w⁢(x),

where C does not depend on α or t. Next, apply Theorem A.1 (b) to conclude that, for every 1<s<∞ and for every v∈R⁢Hs′⁢(w),

∫ℝnGα⁢(x,t)s0s⁢v⁢(x)⁢𝑑w⁢(x)≤C⁢αn⁢r⁢(1s-s0s)⁢∫ℝnG⁢(x,t)s0s⁢v⁢(x)⁢𝑑w⁢(x),

where C does not depend on α or t and where 1≤s0<∞ is arbitrary. From this, given v∈R⁢Hs′⁢(w), 1<q≤s<∞, we can conclude (A.3) by taking s0=sq. ∎

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Received: 2016-12-14

Revised: 2017-06-27

Accepted: 2017-10-19

Published Online: 2017-12-08

Published in Print: 2020-01-01