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for all *< g* ≤ *p* ≤ ∞, and the differential operator *P* (*D*) is obtained from the polynomial *P* (*x*) by substituting

are given.

*ɣ*and Φ

_{1}be nondecreasing and nonnegative functions defined on [0, ∞), and Φ

_{2}is an

*N*-function,

*u, v*and

*w*are weights. A unified version of weighted weak type inequality of the form

for martingale maximal operators *f*
^{∗} is considered, some necessary and su@cient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.

This paper is concerned with the existence of weak solutions for obstacle problems. By means of the Young measure theory and a theorem of Kinderlehrer and Stampacchia, we obtain the needed result.

During the last decade, a number of explicit results about the distributions of exponential functionals of Brownian motion with drift:

In the present paper, we rely extensively on these results to show the existence of limiting measures as

Although a large number of similar studies have been made for, say, one-dimensional diffusions, the present study, which focuses upon Brownian exponential functionals, appears to be new.

We study the effect on sections of a soluble-by-finite group *G* of finite rank of an almost fixed-point-free automorphism *φ* of *G* of finite order. We also elucidate the structure of *G* if *φ* has order 4 and if *G* is also (torsion-free)-by-finite. The latter extends recent work of Xu, Zhou and Liu.

In this paper, we investigate the uniqueness of algebroid functions in angular domain by the method of conformal mapping. We discuss the relations between the Borel directions and uniquenss with the multiple values of algebroid functions and obtain some results which extend some uniqueness results of meromorphic functions to that of algebroid functions.

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

Let *N* be a sufficiently large integer. In this paper, it is proved that, with at most *O*(*N*
^{119/270+}
*
^{s}
*) exceptions, all even positive integers up to

*N*can be represented in the form

where *p*
_{1}
*, p*
_{2}
*, p*
_{3}
*, p*
_{4}
*, p*
_{5}
*, p*
_{6} are prime numbers.

This paper is concerned with the existence of solutions to a class of p(x)-Kirchhoff-type equations with Robin boundary data as follows:

Where

The major aim of the note is to give new brief proofs of the results in the paper “The influence of weakly *H* -subgroups on the structure of finite groups” [Studia Scientiarum Mathematicarum Hungarica, 51 (1), 27–40 (2014)].