For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2)...For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2),Weak L^1(R^2).This extends the theorem of Stein & Wainger and the theo- rem of Weinberg.展开更多
We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas ...We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.展开更多
Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,...Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,q（X） with p, q ￡ （0, ∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.： Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ￡ [1, ∞） and q ￡ [l,∞）, was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].展开更多
文摘For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2),Weak L^1(R^2).This extends the theorem of Stein & Wainger and the theo- rem of Weinberg.
基金supported by Simons Foundation(Grant No.580911(Stinga))Ministerio de Economía y Competitividad/Fondo Europeo de Desarrollo Regional from Government of Spain(Grant No.MTM2015-66157-C2-1-P(Torrea))。
文摘We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.
基金The second author is supported by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China （Grant No. 10XNF090） the third author is supported by National Natural Science Foundation of China （Grant No. 10871025）
文摘Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,q（X） with p, q ￡ （0, ∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.： Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ￡ [1, ∞） and q ￡ [l,∞）, was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].
