Endpoint estimates for the circular maximal function

We consider the problem of endpoint estimates for the circular maximal function defined by

where is the normalized surface area measure on . Let be the closed triangle with vertices . We prove that for , there is a constant such that Furthermore, .

     

An optimal Poincaré inequality in for convex domains

For convex domains with diameter we prove

for any with zero mean value on . We also show that the constant in this inequality is optimal.

     

On the interpolation constant for Orlicz spaces

In this paper we deal with the interpolation from Lebesgue spaces and , into an Orlicz space , where and for some concave function , with special attention to the interpolation constant . For a bounded linear operator in and , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,

where the interpolation constant depends only on and . We give estimates for , which imply . Moreover, if either or , then . If , then , and, in particular, for the case this gives the classical Orlicz interpolation theorem with the constant .

     

Convex curves, Radon transforms and convolution operators defined by singular measures

Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents.     

Global existence for the critical generalized KdV equation

We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation,

The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space , i.e. solutions corresponding to data , 3/4$">, with , where is the solitary wave solution of the equation.

     

Trudinger type inequalities in and their best exponents

We study Trudinger type inequalities in and their best exponents . We show for , ( is the surface area of the unit sphere in ), there exists a constant such that

for all . Here is defined by

It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.

     

The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff's range

Let , , be the Kakeya (Nikodým) maximal operator defined as the supremum of averages over tubes of eccentricity . The (so-called) Fefferman-Stein type inequality:

is shown in the range , where and are some constants depending only on and the dimension and is a weight. The result is a sharp bound up to -factors.

     

The Fefferman-Stein type inequality for the Kakeya maximal operator

Let , , be the Kakeya maximal operator defined as the supremum of averages over tubes of the eccentricity . We shall prove the so-called Fefferman-Stein type inequality for ,

in the range , , with some constants and independent of and the weight .

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

On the local Hölder continuity of the inverse of the -Laplace operator

We prove an interpolation type inequality between , and spaces and use it to establish the local Hölder continuity of the inverse of the -Laplace operator: , for any and in a bounded set in .

     

Blaschke representation of functions on the circle

We prove that every unimodularly bounded measurable function on the complex unit circle admits a representation

where and extend holomorphically into the interior and the exterior of the circle, respectively, vanishes at infinity, and both functions are unimodularly bounded. The representation is unique if .

     

On Korenblum's maximum principle

Let be the Bergman space over the open unit disk in the complex plane. Korenblum's maximum principle states that there is an absolute constant , such that whenever ( ) in the annulus , then . In this paper we prove that Korenblum's maximum principle holds with .

     

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Let be an ideal of over a  -finite measure space , and let stand for the order dual of . For a real Banach space let be a subspace of the space of -equivalence classes of strongly -measurable functions and consisting of all those for which the scalar function belongs to . For a real Banach space a linear operator is said to be order-weakly compact whenever for each the set is relatively weakly compact in . In this paper we examine order-weakly compact operators . We give a characterization of an order-weakly compact operator in terms of the continuity of the conjugate operator of with respect to some weak topologies. It is shown that if is an order continuous Banach function space, is a Banach space containing no isomorphic copy of and is a weakly sequentially complete Banach space, then every continuous linear operator is order-weakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is order-weakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is order-weakly compact iff is reflexive.

     

Integral representation for a class of vector valued operators

Let be a compact space and let , be a (real, for simplicity) Banach space. We consider the space of all continuous -valued functions on , with the supremum norm .

We prove in this paper a Bochner integral representation theorem for bounded linear operators

which satisfy the following condition:

where is the conjugate space of . In the particular case where , this condition is obviously satisfied by every bounded linear operator

and the result reduces to the classical Riesz representation theorem.

If the dimension of is greater than , we show by a simple example that not every bounded linear admits an integral representation of the type above, proving that the situation is different from the one dimensional case.

Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure.

     

Imaginary powers of Laplace operators

We show that if is a second-order uniformly elliptic operator in divergence form on , then . We also prove that the upper bounds remain true for any operator with the finite speed propagation property.

     

Spectral gap for hyperbounded operators

Let be a probability space, and a symmetric linear contraction operator on with and . We prove that is the optimal sufficient condition for to have a spectral gap. Moreover, the optimal sufficient conditions are obtained, respectively, for the defective log-Sobolev and for the defective Poincaré inequality to imply the existence of a spectral gap. Finally, we construct a symmetric, hyperbounded, ergodic contraction -semigroup without a spectral gap.

     

Compact operators whose real and imaginary parts are positive

Let be a compact operator on a Hilbert space such that the operators and are positive. Let be the singular values of and the eigenvalues of , all enumerated in decreasing order. We show that the sequence is majorised by . An important consequence is that, when is less than or equal to , and when this inequality is reversed.

     

A mixed Hölder and Minkowski inequality

Hölder's inequality states that for any with . In the same situation we prove the following stronger chains of inequalities, where :

A similar result holds for complex valued functions with Re substituting for . We obtain these inequalities from some stronger (though slightly more involved) ones.

     

Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

For the classical Hardy-Littlewood maximal function , a well known and important estimate due to Herz and Stein gives the equivalence . In the present note, we study the validity of analogous estimates for maximal operators of the form

where denotes the Lorentz space -norm.