Hengfei ding, changpin li, yangquan chen shanghai university university of california, merced international symposium on fractional pdes. A renormalized riesz potential 3 brdenoting the ball of radius r. The potential theory has been studied very much, especially after the 1 researches of gauss in 1840, where he studied important problems and methods which gave yet remained partly as basic ideas of modern researchs in this. Some embedding theorems and properties of riesz potentials. Note that when and riesz potential coincides with the classical newtonian potential and for satisfies a polyharmonic equation in cf.
Riesz potentials and integral geometry in the space of. The riesz decomposition theorem in potential theory. When and, the riesz potential is a superharmonic function on the entire space. Among the general properties of riesz potentials the following are the most important. Adams abstract this is a survey of various applications of the notion of the choquet integral to questions in potential theory, i. Sobolevs inequality for riesz potentials of functions in.
Riesz introduced three nd integral potential operators to calculate explicitly the potential for hyperbolic, elliptic and parabolic cauchy problems. Its long history in one dimension and some of its recent. Riesz potential on the heisenberg group and modified. Bmo space if the inline pdf is not rendering correctly, you can download the pdf file here. Some properties of the riesz potentials in dunkl analysis. Potential theory of signed riesz kernels 939 the main contribution of this paper is the discovery of an interesting special behaviour of. In this paper, we established the boundedness of i on the padic generalized morrey spaces, as well as the boundedness of the commutators generated by the padic riesz potential i and padic generalized campanato functions. A natural step in the theory of functions spaces was to study orliczmorrey spaces. Riesz and markov processes dynkins characteristic operator. In this paper we study the properties of riesz potential of a function. Second, we give for 1 lq boundedness of the riesz potentials with su. Characterizations for the riesz potential and its commutators on. The following is called the riesz representation theorem. For precise statements, see subharmonic function where it is called the riesz local representation theorem, and riesz theorem where it is simply called the.
This is a survey of various applications of the notion of the cho quet integral to questions in potential theory, i. The topic of this note is the riesz decomposition of excessive functions for a nice strong markov process x. Some embedding theorems and properties of riesz potentials p k b a r fx k b a r. Riesz potential with lorentz distance hyperbolic riesz potentials 555 28. Endowed with a proper normalization, and regarded as a limit of riesz potentials, it is equal to diracs distribution however, a different normalization turns the nonlinear operator. Not all such potentials, however, are admissible cf. Blakely, potential theory in gravity a nd magnetic applications cambridge university pr ess, 1996 4 inan c. In classical potential theory in the complex plane, there is a notion of potential, more general than logarithmic potential, that yields a riesz decomposition of all admissible superharmonic functions.
A renormalized riesz potential and applications semantic. Theorem 1 if t is a bounded linear functional on a hilbert space h then there exists some g. An essential role in the question of convergence of these truncations is played by the assertion established in 996. In dunkl theory on rd which generalizes classical fourier analysis, we study. In newtons theory of gravitation the potential of a ball br0. In mathematics, the riesz potential is a potential named after its discoverer, the hungarian mathematician marcel riesz. Riesz decomposition theorem encyclopedia of mathematics. This paper offers a new perspective to look at the riesz potential. For all the riesz potential is a lower semicontinuous function on, continuous outside. Roughly speaking, this asserts that a super or subharmonic function is the sum of a potential and a harmonic function. The riesz representation theorem ma 466 kurt bryan let h be a hilbert space over lr or cl, and t a bounded linear functional on h a bounded operator from h to the. That was my rst time to teach such a course and, i have to say, besides sporadic knowledge of a few facts directly related to complex analysis, i had no serious knowledge of the subject. Turesson, nonlinear potential theory and weighted sobolev spaces, lecture notes in math. In a sense, the riesz potential defines an inverse for a power of the laplace operator on euclidean space.
Symmetry group classification and conservation laws of the. Di erentiability properties of riesz potentials of nite. Classical potential theory attended by an excellent class of graduate students of the department of mathematics of washington university. Due to this reason, many interesting works were devoted to the theory of harmonic analysis on hn in 1015 and the references therein. In this paper we study the fractional maximal operator m. For the theory on triebellizorkin spaces we refer to 12, 9 and 6. On the role of riesz potentials in poissons equation and sobolev. The riesz decomposition theorem in potential theory david singman the classical version of this theorem says that any positive nonconstant superharmonic function on an open subset. In this section we record various background results on lp, riesz potential, and triebellizorkin spaces. Necessary and sufficient conditions for the boundedness of. Linear potentials in nonlinear potential theory 3 finally, a unifying approach allowing to view both estimate 1. Spanne and adams type boundedness of the riesz potential on the. Except for the divergence theorem, the arzelaascoli theorem, the radonriesz representa. In the plane we consider the logarithmic potential of a.
Commutator of riesz potential in p adic generalized. High order numerical methods for the riesz derivatives and. On limiting trace inequalities for vectorial differential operators. In our study the most interesting is the weak singular integral operator i. In section 4 we compute the exponential transform in some simple cases, mainly for. Research article riesz potential on the heisenberg group. Thus, what we call the riesz representation theorem is stated in three parts as theorems 2. As application, we prove a weighted generalized sobolev. The operator of riesz potential type 2295 and since the function s 7. High order numerical methods for the riesz derivatives and the space riesz fractional differential equation. They generalize to several variables the riemannliouville integrals of one variable. Adams, choquet integrals in potential theory, publ. When, the riesz potential is a subharmonic function outside.
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