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Register a new userOn the Bounds of Weak $(1,1)$ Norm of Hardy-Littlewood Maximal Operator with $Llog L({mathbb S^{n-1}})$ Kernels
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Let $Omegain L^1{({mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_Omega$ be the Hardy-Littlewood maximal operator associated with $Omega$ defined by $M_Omega(f)(x) = sup_{r>0}frac1{r^n}int_{|x-y|<r}|Omega(x-y)f(y)|dy.$ It was shown by Christ and Rubio de Francia that $|M_Omega(f)|_{L^{1,infty}({mathbb R^n})} le C(|Omega|_{Llog L({mathbb S^{n-1}})}+1)|f|_{L^1({mathbb R^n})}$ provided $Omegain Llog L {({mathbb S^{n-1}})}$. In this paper, we show that, if $Omegain Llog L({mathbb S^{n-1}})$, then for all $fin L^1({mathbb R^n})$, $M_Omega$ enjoys the limiting weak-type behaviors that $$lim_{lambdato 0^+}lambda|{xin{mathbb R^n}:M_Omega(f)(x)>lambda}| = n^{-1}|Omega|_{L^1({mathbb S^{n-1}})}|f|_{L^1({mathbb R^n})}.$$ This removes the smoothness restrictions on the kernel $Omega$, such as Dini-type conditions, in previous results. To prove our result, we present a new upper bound of $|M_Omega|_{L^1to L^{1,infty}}$, which essentially improves the upper bound $C(|Omega|_{Llog L({mathbb S^{n-1}})}+1)$ given by Christ and Rubio de Francia. As a consequence, the upper and lower bounds of $|M_Omega|_{L^1to L^{1,infty}}$ are obtained for $Omegain Llog L {({mathbb S^{n-1}})}$.
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Let $Omega$ be a function of homogeneous of degree zero and vanish on the unit sphere $mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_Omega$ associated with rough kernel $Omega$. We show that, if $Omegain Llog L(mathbb S^{n})$, then $lim_{lambdato0^+}lambda|{xinmathbb{R}^n:|T_Omega(f)(x)|>lambda}| = n^{-1}|Omega|_{L^1(mathbb {S}^n)}|f|_{L^1(mathbb{R}^n)},quad0le fin L^1(mathbb{R}^n).$ Moreover,$(n^{-1}|Omega|_{L^1(mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_Omega$ when $Omegain Llog L(mathbb{S}^{n-1})$. Corresponding results for rough bilinear singular integral operators defined in the form $T_{vecOmega}(f_1,f_2) = T_{Omega_1}(f_1)cdot T_{Omega_2}(f_2)$ have also been established.
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.
In this note besides two abstra
This paper shows $$ sup_{fin H^s(mathbb{R}^n)}dim _Hleft{xinmathbb{R}^n: lim_{trightarrow0}e^{it(-Delta)^alpha}f(x) eq f(x)right}leq n+1-frac{2(n+1)s}{n} text{under} begin{cases} ngeq2; alpha>frac12; frac{n}{2(n+1)}<sleqfrac{n}{2} . end{cases} $$
Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition and $L$ a one to one operator of type $omega$ in $L^2({mathbb R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W!H_L^{p(cdot)}(mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W!T^{p(cdot)}(mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $ abla L^{-1/2}$ is bounded from $W!H_L^{p(cdot)}(mathbb R^n)$ to the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of ${e^{-tL}}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.
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