{"history":[],"article":{"id":"c8f93ee2-5d1b-4830-8900-9548fad64ad6","title":"Riesz potential","slug":"riesz-potential","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Riesz [potential](https://bliptext.com/articles/potential-theory)** is a [potential](https://bliptext.com/articles/potential-theory) named after its discoverer, the Hungarian mathematician [Marcel Riesz](https://bliptext.com/articles/marcel-riesz).\nIn a sense, the Riesz potential defines an inverse for a power of the [Laplace operator](https://bliptext.com/articles/laplace-operator) on Euclidean space.\nThey generalize to several variables the [Riemann–Liouville integrals](https://bliptext.com/articles/riemann-liouville-integral) of one variable.\n\n# Definition\nIf 0 < α < n, then the **R**iesz potential Iαf of a [locally integrable function](https://bliptext.com/articles/locally-integrable-function) f on **R**n is the function defined by\nwhere the constant is given by\nThis [singular integral](https://bliptext.com/articles/singular-integral) is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(**R**n) with 1 ≤ p < n/α.\nIn fact, for any 1 ≤ p (p>1 is classical, due to Sobolev, while for p=1 see, the rate of decay of f and that of Iαf are related in the form of an inequality (the [Hardy–Littlewood–Sobolev inequality](https://bliptext.com/articles/hardy-littlewood-sobolev-inequality)) where Rf=DI_1f is the vector-valued [Riesz transform](https://bliptext.com/articles/riesz-transform). More generally, the operators Iα are well-defined for [complex](https://bliptext.com/articles/complex-number) α such that 0 < Re α < n.\nThe Riesz potential can be defined more generally in a [weak sense](https://bliptext.com/articles/distribution-mathematics) as the [convolution](https://bliptext.com/articles/convolution)\nwhere Kα is the locally integrable function:\nThe Riesz potential can therefore be defined whenever f is a compactly supported distribution.\nIn this connection, the **R**iesz potential of a positive [Borel measure](https://bliptext.com/articles/borel-measure) μ with [compact support](https://bliptext.com/articles/support-measure-theory) is chiefly of interest in [potential theory](https://bliptext.com/articles/potential-theory) because Iαμ is then a (continuous) [subharmonic function](https://bliptext.com/articles/subharmonic-function) off the support of μ, and is [lower semicontinuous](https://bliptext.com/articles/lower-semicontinuous) on all of **R**n.\nConsideration of the [Fourier transform](https://bliptext.com/articles/fourier-transform) reveals that the Riesz potential is a [Fourier multiplier](https://bliptext.com/articles/fourier-multiplier).\nIn fact, one has\nand so, by the [convolution theorem](https://bliptext.com/articles/convolution-theorem),\nThe Riesz potentials satisfy the following [semigroup](https://bliptext.com/articles/semigroup) property on, for instance, [rapidly decreasing](https://bliptext.com/articles/rapidly-decreasing) [continuous functions](https://bliptext.com/articles/continuous-function)\nprovided\nFurthermore, if 0 < Re α < n–2, then\nOne also has, for this class of functions,","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T08:21:42.973Z","updated_at":"2025-01-26T08:21:42.973Z","search_vector":"'potenti':2 'riesz':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Riesz [potential](https://bliptext.com/articles/potential-theory)** is a [potential](https://bliptext.com/articles/potential-theory) named after its discoverer, the Hungarian mathematician [Marcel Riesz](https://bliptext.com/articles/marcel-riesz).\nIn a sense, the Riesz potential defines an inverse for a power of the [Laplace operator](https://bliptext.com/articles/laplace-operator) on Euclidean space.\nThey generalize to several variables the [Riemann–Liouville integrals](https://bliptext.com/articles/riemann-liouville-integral) of one variable.\n\n# Definition\nIf 0 1 is classical, due to Sobolev, while for p=1 see, the rate of decay of f and that of Iαf are related in the form of an inequality (the [Hardy–Littlewood–Sobolev inequality](https://bliptext.com/articles/hardy-littlewood-sobolev-inequality)) where Rf=DI_1f is the vector-valued [Riesz transform](https://bliptext.com/articles/riesz-transform). More generally, the operators Iα are well-defined for [complex](https://bliptext.com/articles/complex-number) α such that 0 < Re α < n.\nThe Riesz potential can be defined more generally in a [weak sense](https://bliptext.com/articles/distribution-mathematics) as the [convolution](https://bliptext.com/articles/convolution)\nwhere Kα is the locally integrable function:\nThe Riesz potential can therefore be defined whenever f is a compactly supported distribution.\nIn this connection, the **R**iesz potential of a positive [Borel measure](https://bliptext.com/articles/borel-measure) μ with [compact support](https://bliptext.com/articles/support-measure-theory) is chiefly of interest in [potential theory](https://bliptext.com/articles/potential-theory) because Iαμ is then a (continuous) [subharmonic function](https://bliptext.com/articles/subharmonic-function) off the support of μ, and is [lower semicontinuous](https://bliptext.com/articles/lower-semicontinuous) on all of **R**n.\nConsideration of the [Fourier transform](https://bliptext.com/articles/fourier-transform) reveals that the Riesz potential is a [Fourier multiplier](https://bliptext.com/articles/fourier-multiplier).\nIn fact, one has\nand so, by the [convolution theorem](https://bliptext.com/articles/convolution-theorem),\nThe Riesz potentials satisfy the following [semigroup](https://bliptext.com/articles/semigroup) property on, for instance, [rapidly decreasing](https://bliptext.com/articles/rapidly-decreasing) [continuous functions](https://bliptext.com/articles/continuous-function)\nprovided\nFurthermore, if 0 < Re α < n–2, then\nOne also has, for this class of functions,","revisionCount":0},"hasMore":false,"nextCursor":null}

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