Complex Analysis and Operator Theory最新文献

Given a positive Borel measure (mu ) on the one-dimensional Euclidean space (textbf{R}), consider the centered Hardy–Littlewood maximal function (M_mu ) acting on a finite positive Borel measure (nu ) by

where (r_0(x) = inf {r> 0: mu (B(x,r)) > 0}) and B(x, r) denotes the closed ball with centre x and radius (r > 0). In this note, we restrict our attention to Radon measures (mu ) on the positive real line ([0,+infty )). We provide a complete characterization of measures having weak-type asymptotic properties for the centered maximal function. Although we don’t know whether this fact can be extended to measures on the entire real line (textbf{R}), we examine some criteria for the existence of the weak-type asymptotic properties for (M_mu ) on (textbf{R}). We also discuss further properties, and compute the value of the relevant asymptotic quantity for several examples of measures.

We consider a bounded strictly pseudoconvex domain (Omega subset mathbb {C}^{n}) with (C^{2}) boundary. Then, we show that any compact Ahlfors–David regular subset of (partial Omega ) of Hausdorff dimension (beta in (0,2n-1]) contains a peak set E of Hausdorff dimension equal to (beta ).

Let (xi in mathbb {R}), and (rho _iin mathbb {R}) with (0<|rho _i|<1) for (1le ile 3). For an expanding real matrix

and an integer digit set (D={(0,0,0)^t, (1,0,0)^t, (0,1,0)^t, (0,0,1)^t }subset mathbb {Z}^3), let (mu _{M,D}) be the self-affine measure defined by (mu _{M,D}(cdot )=frac{1}{|D|}sum _{din D}mu _{M,D}(M(cdot )-d)). In this paper, we prove that if (rho _1=rho _2), then (L^2(mu _{M,D})) admits an infinite orthogonal set of exponential functions if and only if (|rho _i|=(p_i/q_i)^{frac{1}{r_i}}) for some (p_i,q_i,r_iin mathbb {N}^+) with (gcd (p_i,q_i)=1) and (2|q_i), (i=1,2). In particular, if (rho _1,rho _2,rho _3in {frac{p}{q}:p,qin 2mathbb {Z}+1}) and (rho _1=rho _2), then there exist at most 4 mutually orthogonal exponential functions in (L^2(mu _{M,D})), and the number 4 is the best.

Let (mathcal {M}) be a von Neumann algebra with a normal faithful semifinite trace. In this paper, we consider that in n-tuples of noncommutative (L_p)-spaces (l_s^{(n)}(L_p(mathcal {M}))), the norm is invariant under the action of invertible elements in (mathcal {M}). Then we prove that the complex interpolating theorem in the case of (l_s^{(n)}(L_p(mathcal {M}))). Using this result, we obtain that Clarkson’s inequalities for n-tuples of operators with weighted norm of noncommutative (L_p)-spaces, where the weight being a positive invertible operator in (mathcal {M}).

In this paper we propose an Almansi-type decomposition for slice regular functions of several quaternionic variables. Our method yields (2^n) distinct and unique decompositions for any slice function with domain in (mathbb {H}^n). Depending on the choice of the decomposition, every component is given explicitly, uniquely determined and exhibits desirable properties, such as harmonicity and circularity in the selected variables. As consequences of these decompositions, we give another proof of Fueter’s Theorem in (mathbb {H}^n), establish the biharmonicity of slice regular functions in every variable and derive mean value and Poisson formulas for them.

We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. Clifford analogues of complex holomorphic functions—called monogenic functions—are defined by means of the Dirac operators that factor a certain wave operator. One of the fundamental features of quaternionic analysis is the invariance of quaternionic analogues of holomorphic function under conformal (or Möbius) transformations. A similar invariance property is known to hold in the context of Clifford algebras associated to positive definite quadratic forms. We generalize these results to the case of Clifford algebras associated to all non-degenerate quadratic forms. This result puts the indefinite signature case on the same footing as the classical positive definite case.

This paper employs Nevanlinna value distribution theory in several complex variables to explore the characteristics and form of finite as well as infinite order solutions of Circular-type nonlinear partial differential-difference equations in (mathbb {C}^n). The results obtained in this paper contribute to the improvement and generalization of some recent results. In addition, illustrative examples are provided to validate the conclusions drawn from the main results. Moreover, we investigate the infinite-order solutions of Circular-type functional equations in ( mathbb {C}^n ).