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Register a new userA weak type estimate for regular fractional sparse operators
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In this note the weak type estimates for fractional integrals are studied. More precisely, we adapt the arguments of Domingo-Salazar, Lacey, and Rey to obtain improvements for the endpoint weak type estimates for regular fractional sparse operators.
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In this note we prove the estimate $M^{sharp}_{0,s}(Tf)(x) le c,M_gamma f(x)$ for general fractional type operators $T$, where $M^{sharp}_{0,s}$ is the local sharp maximal function and $M_gamma$ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of $Tf$ by $M_gamma f$. Similar estimates hold for $T$ replaced by fractional type operators with kernels satisfying H{o}rmander-type conditions or integral operators with homogeneous kernels, and $M_gamma $ replaced by an appropriate maximal function $M_T$. We also prove two-weight, $L^p_v$-$L^q_w$ estimates for the fractional type operators described above for $1<p< q<infty$ and a range of $q$. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.
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 $Omega_1,Omega_2$ be functions of homogeneous of degree $0$ and $vecOmega=(Omega_1,Omega_2)in Llog L(mathbb{S}^{n-1})times Llog L(mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{vecOmega}$ and bilinear singular integral $T_{vecOmega}$ associated with rough kernel $vecOmega$. For all $f,gin L^1(mathbb{R}^n)$, we show that $$lim_{lambdato 0^+}lambda |big{ xinmathbb{R}^n:M_{vecOmega}(f_1,f_2)(x)>lambdabig}|^2 = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{omega_{n-1}^2}prodlimits_{i=1}^2| f_i|_{L^1}$$ and $$lim_{lambdato 0^+}lambda|big{ xinmathbb{R}^n:| T_{vecOmega}(f_1,f_2)(x)|>lambdabig}|^{2} = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{n^2}prodlimits_{i=1}^2| f_i|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{vecOmega}$ and $T_{vecOmega}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.
We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by multiparameter sparse operators associated with the corresponding dyadic grid. We also obtain supplementary results about the absence of sparse domination for the strong dyadic maximal function.
It is well known that the weak ($1,1$) bounds doesnt hold for the strong maximal operators, but it still enjoys certain weak $Llog L$ type norm inequality. Let $Phi_n(t)=t(1+(log^+t)^{n-1})$ and the space $L_{Phi_n}({mathbb R^{n}})$ be the set of all measurable functions on ${mathbb R^{n}}$ such that $|f|_{L_{Phi_n}({mathbb R^{n}})} :=|Phi_n(|f|)|_{L^1({mathbb R^{n}})}<infty$. In this paper, we introduce a new weak norm space $L_{Phi_n}^{1,infty}({mathbb R^{n}})$, which is more larger than $L^{1,infty}({mathbb R^{n}})$ space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that $ max{{2^n}{((n-1)!)^{-1}},1}$ is a lower bound for the best constant of the $L_{Phi_n}to L_{Phi_n}^{1,infty}$ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.
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