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Register a new userA sparse domination for the Marcinkiewicz integral with rough kernel and applications
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Let $Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $mu_{Omega}$ be the higher-dimensional Marcinkiewicz integral defined by $$mu_Omega(f)(x)= Big(int_0^inftyBig|int_{|x-y|leq t}frac{Omega(x-y)}{|x-y|^{n-1}}f(y)dyBig|^2frac{dt}{t^3}Big)^{1/2}. $$ In this paper, the authors establish a bilinear sparse domination for $mu_{Omega}$ under the assumption $Omegain L^{infty}(S^{n-1})$. As applications, some quantitative weighted bounds for $mu_{Omega}$ are obtained.
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We present a general approach to sparse domination based on single-scale $L^p$-improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ-Hytonen-Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderon-Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of $mathbb R^n$.
We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmunds $L{(log L)^{{frac12}}}$ inequality is the core of our lacunary multi-frequency projection proof. We use the Walsh setting to avoid extra complications in the arguments.
The purpose of this paper is to study the sparse bound of the operator of the form $f mapsto psi(x) int f(gamma_t(x))K(t)dt$, where $gamma_t(x)$ is a $C^infty$ function defined on a neighborhood of the origin in $(x, t) in mathbb R^n times mathbb R^k$, satisfying $gamma_0(x) equiv x$, $psi$ is a $C^infty$ cut-off function supported on a small neighborhood of $0 in mathbb R^n$ and $K$ is a Calderon-Zygmund kernel suppported on a small neighborhood of $0 in mathbb R^k$. Christ, Nagel, Stein and Wainger gave conditions on $gamma$ under which $T: L^p mapsto L^p (1<p<infty)$ is bounded. Under the these same conditions, we prove sparse bounds for $T$, which strengthens their result. As a corollary, we derive weighted norm estimates for such operators.
By a reduction method, the limiting weak-type behaviors of factional maximal operators and fractional integrals are established without any smoothness assumption on the kernel, which essentially improve and extend previous results. As a byproduct, we characterize the boundedness of several operators by the membership of their kernel in Lebesgue space on sphere.
Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_{Omega}$ be the homogeneous singular integral operator with kernel $frac{Omega(x)}{|x|^d}$ and $T_{Omega,,b}$ be the commutator of $T_{Omega}$ with symbol $b$. In this paper, we prove that if $Omegain L(log L)^2(S^{d-1})$, then for $bin {rm BMO}(mathbb{R}^d)$, $T_{Omega,,b}$ satisfies an endpoint estimate of $Llog L$ type.
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