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. Riesz and markov processes dynkins characteristic operator. High order numerical methods for the riesz derivatives and. Riesz potential on the heisenberg group and modified. Adams, choquet integrals in potential theory, publ. Roughly speaking, this asserts that a super or subharmonic function is the sum of a potential and a harmonic function. 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. Second, we give for 1 lq boundedness of the riesz potentials with su. In our study the most interesting is the weak singular integral operator i. A renormalized riesz potential and applications semantic. Some properties of the riesz potentials in dunkl analysis. For the theory on triebellizorkin spaces we refer to 12, 9 and 6.
Riesz potentials and integral geometry in the space of. In dunkl theory on rd which generalizes classical fourier analysis, we study. Necessary and sufficient conditions for the boundedness of. When and, the riesz potential is a superharmonic function on the entire space. Spanne and adams type boundedness of the riesz potential on the. They generalize to several variables the riemannliouville integrals of one variable. Some embedding theorems and properties of riesz potentials p k b a r fx k b a r. For all the riesz potential is a lower semicontinuous function on, continuous outside.
Hengfei ding, changpin li, yangquan chen shanghai university university of california, merced international symposium on fractional pdes. Riesz potential with lorentz distance hyperbolic riesz potentials 555 28. When, the riesz potential is a subharmonic function outside. On limiting trace inequalities for vectorial differential operators. 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. In mathematics, the riesz potential is a potential named after its discoverer, the hungarian mathematician marcel riesz. In the plane we consider the logarithmic potential of a. The following is called the riesz representation theorem. In this section we record various background results on lp, riesz potential, and triebellizorkin spaces. This is a survey of various applications of the notion of the cho quet integral to questions in potential theory, i.
Di erentiability properties of riesz potentials of nite. Among the general properties of riesz potentials the following are the most important. As application, we prove a weighted generalized sobolev. In a sense, the riesz potential defines an inverse for a power of the laplace operator on euclidean space. Characterizations for the riesz potential and its commutators on. This paper offers a new perspective to look at the riesz potential. For precise statements, see subharmonic function where it is called the riesz local representation theorem, and riesz theorem where it is simply called the. 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. Note that when and riesz potential coincides with the classical newtonian potential and for satisfies a polyharmonic equation in cf.
In this paper we study the fractional maximal operator m. Turesson, nonlinear potential theory and weighted sobolev spaces, lecture notes in math. Symmetry group classification and conservation laws of the. The potential theory has been studied very much, especially after the 1. The potential satisfies in the distributional sense is the characteristic function of the set. In this paper we study the properties of riesz potential of a function. Thus, what we call the riesz representation theorem is stated in three parts as theorems 2. Some embedding theorems and properties of riesz potentials. A renormalized riesz potential 3 brdenoting the ball of radius r. In section 4 we compute the exponential transform in some simple cases, mainly for. Its long history in one dimension and some of its recent.
Riesz introduced three nd integral potential operators to calculate explicitly the potential for hyperbolic, elliptic and parabolic cauchy problems. Adams abstract this is a survey of various applications of the notion of the choquet integral to questions in potential theory, i. On isoperimetric inequalities of riesz potentials and. Riesz decomposition theorem for super or subharmonic functions. Classical potential theory attended by an excellent class of graduate students of the department of mathematics of washington university. Except for the divergence theorem, the arzelaascoli theorem, the radonriesz representa. Riesz decomposition theorem encyclopedia of mathematics. Linear potentials in nonlinear potential theory 3 finally, a unifying approach allowing to view both estimate 1. The riesz decomposition theorem in potential theory. Research article riesz potential on the heisenberg group.
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. Bmo space if the inline pdf is not rendering correctly, you can download the pdf file here. Commutator of riesz potential in p adic generalized. On the role of riesz potentials in poissons equation and sobolev. An essential role in the question of convergence of these truncations is played by the assertion established in 996. Theorem 1 if t is a bounded linear functional on a hilbert space h then there exists some g. Blakely, potential theory in gravity a nd magnetic applications cambridge university pr ess, 1996 4 inan c. In newtons theory of gravitation the potential of a ball br0. 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. A natural step in the theory of functions spaces was to study orliczmorrey spaces. High order numerical methods for the riesz derivatives and the space riesz fractional differential equation. 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. The operator of riesz potential type 2295 and since the function s 7.
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