A note on maximal operator on ℓ{pn} and Lp(x)(ℝ)

Let $ v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform \operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilon f(x-yv(x)) \frac{dy}y where $ \epsilon $ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.\thinspace M.\thinspace Stein, that if $ v$ is Lipschitz, there is a positive $ \epsilon $ for which the transform above is bounded on $ L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $ v$. This sufficient condition is that this new maximal function be bounded on some $ L ^{p}$, for some $ 1<p<2$. We show that the maximal function is bounded from $ L ^{2}$ to weak $ L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored.

On certain maximal functions and approach regions

Maximal operator on variable Lebesgue spaces with radial exponent

Hardy-Littlewood maximal operators defined forX-valued functions on ℝn,X a Kothe function space, are considered. Necessary and sufficient conditions on weightsv(x) for the existence of a weightu(x) such that the maximal operator is bounded fromLxp(v) toLxp(u) are given. The cases ofX=lr andX=ls,r are studied in detail. Also, the behavior of the weights under convexity assumptions onX is considered.

It is shown that the stationary, autoregressive, Markovian minification processes introduced by Tavares and Sim can be extended to give processes with marginal distributions other than the exponential and Weibull distributions. Necessary and sufficient conditions on the hazard rate of the marginal distributions are given for a minification process to exist. Results are given for the derivation of the autocorrelation function; these correct the expression for the Weibull given by Sim. Monotonic transformations of the minification processes are also discussed and generate a whole new class of autoregressive processes with fixed marginal distributions. Stationary processes generated by a maximum operation are also introduced and a comparison of three different Markovian processes with uniform marginal distributions is given.

A weighted theory for multilinear fractional integral operators and maximal functions is presented. Sufficient conditions for the two weight inequalities of these operators are found, including “power and logarithmic bumps” and anA ∞ condition. For one weight inequalities a necessary and sufficient condition is then obtained as a consequence of the two weight inequalities. As an application, Poincaré and Sobolev inequalities adapted to the multilinear setting are presented.

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,\mu)$. The embedding of the Newton-Morrey-Sobolev space into the H\"older space is obtained if $\mathscr{X}$ supports a weak Poincar\'e inequality and the measure $\mu$ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors $Q$-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Haj{\l}asz gradient, the authors also introduce the Haj{\l}asz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Haj{\l}asz-Morrey-Sobolev space when $\mu$ is doubling and $\mathscr{X}$ supports a weak Poincar\'e inequality. In particular, on the Euclidean space ${\mathbb R}^n$, the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Haj{\l}asz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when $(\mathscr{X},d)$ is geometrically doubling and $\mu$ a non-negative Radon measure, the boundedness of some modified (fractional) maximal operators on modified Morrey spaces is presented; as an application, when $\mu$ is doubling and satisfies some measure decay property, the authors further obtain the boundedness of some (fractional) maximal operators on Morrey spaces, Newton-Morrey-Sobolev spaces and Haj{\l}asz-Morrey-Sobolev spaces.

Let 0 ≤ α &lt; n , M α be the fractional maximal operator, M ♯ be the sharp maximal operator and b be the locally integrable function. Denote by [ b , M α ] and [ b , M ♯ ] be the commutators of the fractional maximal operator M α and the sharp maximal operator M ♯ . In this paper, we show some necessary and sufficient conditions for the boundedness of the commutators [ b , M α ] and [ b , M ♯ ] on slice spaces when the function b is the Lipschitz function, by which some new characterizations of the non-negative Lipschitz function are obtained.

In a filtered measure space, a necessary and sufficient Sawyer type condition is given under which a positive operator is bounded from $$L^1(d\mu )$$ to $$L^{q,\infty }(d\mu )$$ with $$1<q <\infty $$ . This result can be recognized as the endpoint case of Tanaka and Terasawa (Arch Math (Basel) 101:559–568, 2013, Theorem 1.1). We also establish a similar weak type inequality for generalized Doob’s maximal operators.

Let X be a metric space with doubling measure, and L be a nonnegative self-adjoint operator on L²(X) such that the semigroup kernels of <roman>e</roman>^-tLsatisfy the upper and lower Gaussian bounds. Given Hörmander-type multipliers m_i, 1≤ i≤ N, with the uniform estimate, we prove an optimal sqrtłog(1+N)bound in L^p for the maximal function sup_1≤ i≤ Nm_i(L)f by making use of the Doob transform and the <roman>exp</roman>(L²) estimate due to Chang-Wilson-Wolff for the dyadic martingale square function. Based on this, we establish a sufficient condition on the function m such that the maximal function mathcal Mm, Lf(x)=sup_t&gt;0m(tL)f(x) is bounded on L^p(X).

The concept of Lebesgue space has been generalized to the grand Lebesgue space with non-weight and weight, and the classical Lorentz space concept has been generalized to grand Lorentz spaces with a similar logic. In this study, instead of using rearrangement for a measurable function, weighted Grand Lorentz spaces are defined by using the maximal function 1&lt;=p, q&lt;=∞ where the weight function is measurable, complex valued, and locally bounded. In addition, multiplication operators on weighted grand Lorentz spaces are defined and the fundamental properties of these operators such as boundedness, closed range, invertibility, compactness, and closedness are characterized.

In this paper, we are first concerned with viscous approximations for the three-dimensional axisymmetric incompressible Euler equations. It is proved that the viscous approximations, which are the solutions of the corresponding Navier-Stokes equations, converge strongly in $$ L^{2} {\left( {{\left[ {0,T} \right]};L^{2}_{{loc}} {\left( {R^{3} } \right)}} \right)} $$ provided that they have strong convergence in the region away from the symmetry axis. This result has been proved by the authors for the approximate solutions generated by smoothing the initial data, with no restriction of the sign of the initial data. Then we discuss the decay rate for maximal vorticity function, which is established for both approximate solutions generated by smoothing the initial data and viscous approximations respectively. One sufficient condition to guarantee the strong convergence in the region away from the symmetry axis is given, and a decay rate for maximal vorticity function in the region away from the symmetry axis is obtained for non-negative initial vorticity.

Well-posedness of semilinear heat equations in L1

We analyze the space-times admitting two shear-free geodesic null congruences. The integrability conditions are presented in a plain tensorial way as equations on the volume element U of the time-like 2-plane that these directions define. From these we easily deduce significant consequences. We obtain explicit expressions for the Ricci and Weyl tensors in terms of U and its first and second order covariant derivatives. We study the different compatible Petrov-Bel types and give the necessary and sufficient conditions that characterize every type in terms of U. The type D case is analyzed in detail and we show that every type D space-time admitting a 2 + 2 conformal Killing tensor also admits a conformal Killing-Yano tensor.

Sufficient conditions are given for a linear differential equation of order n to have a solution which behaves asymptotically like a given polynomial of degree $ < n$. The integral smallness conditions on the coefficient and forcing functions are stated largely in terms of ordinary (rather than absolute) convergence, and the manner in which the solution behaves like the given polynomial is specified precisely.