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Register a new userThe limiting weak type behaviors and The lower bound for a new weak $Llog L$ type norm of strong maximal operators
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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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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.
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.
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 $Omegain Llog L {({mathbb S^{n-1}})}$.
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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