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Register a new userBilinear Wavelet Representation of Calderon-Zygmund Forms
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We represent a bilinear Calderon-Zygmund operator at a given smoothness level as a finite sum of cancellative, complexity zero operators, involving smooth wavelet forms, and continuous paraproduct forms. This representation results in a sparse $T(1)$-type bound, which in turn yields directly new sharp weighted bilinear estimates on Lebesgue and Sobolev spaces. Moreover, we apply the representation theorem to study fractional differentiation of bilinear operators, establishing Leibniz-type rules in weighted Sobolev spaces which are new even in the simplest case of the pointwise product.
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This article develops a novel approach to the representation of singular integral operators of Calderon-Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderon-Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of $T(1)$ theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the $A_2$ theorem; that is, sharp dependence of the Sobolev norm of $T$ on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative $A_p$ estimates which are best known, and sharp in the range $max{p,p}geq 3$ for the fully cancellative case.
We study interior $L^p$-regularity theory, also known as Calderon-Zygmund theory, of the equation [ int_{mathbb{R}^n} int_{mathbb{R}^n} frac{K(x,y) (u(x)-u(y)), (varphi(x)-varphi(y))}{|x-y|^{n+2s}}, dx, dy = langle f, varphi rangle quad varphi in C_c^infty(mathbb{R}^n). ] For $s in (0,1)$, $t in [s,2s]$, $p in [2,infty)$, $K$ an elliptic, symmetric, Holder continuous kernel, if $f in left (H^{t,p}_{00}(Omega)right )^ast$, then the solution $u$ belongs to $H^{2s-t,p}_{loc}(Omega)$ as long as $2s-t < 1$. The increase in differentiability is independent of the Holder coefficient of $K$. For example, our result shows that if $fin L^{p}_{loc}$ then $uin H^{2s-delta,p}_{loc}$ for any $deltain (0, s]$ as long as $2s-delta < 1$. This is different than the classical analogue of divergence-form equations ${rm div}(bar{K} abla u) = f$ (i.e. $s=1$) where a $C^gamma$-Holder continuous coefficient $bar{K}$ only allows for estimates of order $H^{1+gamma}$. In fact, it is another appearance of the differential stability effect observed in many forms by many authors for this kind of nonlocal equations -- only that in our case we do not get a small differentiability improvement, but all the way up to $min{2s-t,1}$. The proof argues by comparison with the (much simpler) equation [ int_{mathbb{R}^n} K(z,z) (-Delta)^{frac{t}{2}} u(z) , (-Delta)^{frac{2s-t}{2}} varphi(z), dz = langle g,varphirangle quad varphi in C_c^infty(mathbb{R}^n). ] and showing that as long as $K$ is Holder continuous and $s,t, 2s-t in (0,1)$ then the commutator [ int_{mathbb{R}^n} K(z,z) (-Delta)^{frac{t}{2}} u(z) , (-Delta)^{frac{2s-t}{2}} varphi(z), dz - cint_{mathbb{R}^n} int_{mathbb{R}^n} frac{K(x,y) (u(x)-u(y)), (varphi(x)-varphi(y))}{|x-y|^{n+2s}}, dx, dy ] behaves like a lower order operator.
Given $2leq p<infty$, $sin (0, 1)$ and $tin (1, 2s)$, we establish interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal equations of the form [ int_{Omega} int_{Omega} Kleft (x,|x-y|,frac{x-y}{|x-y|}right ) frac{(u(x)-u(y))(varphi(x)-varphi(y))}{|x-y|^{n+2s}} dx dy = g[varphi], quad forall phiin C_c^{infty}(Omega) ] where $Omegasubset mathbb{R}^{n}$ is an open set. Here we assume $K$ is bounded, nonnegative and continuous in the first entry -- and ellipticity is ensured by assuming that $K$ is strictly positive in a cone. The setup is chosen so that it is applicable for nonlocal equations on manifolds, but the structure of the equation is general enough that it also applies to the certain fractional $p$-Laplace equations around points where $u in C^1$ and $| abla u| eq 0$.
We deal with a global Calderon-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1<p le 2-frac{1}{n}$. In addition, the emphasis of this paper is in obtaining the Lorentz bounds for the gradient of solutions with the use of fractional maximal operators.
In this paper, we determine the $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness of the bilinear Hilbert transform $H_{gamma}(f,g)$ along a convex curve $gamma$ $$H_{gamma}(f,g)(x):=mathrm{p.,v.}int_{-infty}^{infty}f(x-t)g(x-gamma(t)) ,frac{textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $frac{1}{p}+frac{1}{q}=frac{1}{r}$, and $r>frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{gamma}(f,g)$ along a convex curve $gamma$ $$M_{gamma}(f,g)(x):=sup_{varepsilon>0}frac{1}{2varepsilon}int_{-varepsilon}^{varepsilon}|f(x-t)g(x-gamma(t))| ,textrm{d}t.$$
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