Quasitransitive relation

In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable.

If 0 < α < n, then the Riesz potential I_αf of a locally integrable function f on Rⁿ is the function defined by

where the constant is given by

$c_{\alpha }={\pi }^{n/2}{2}^{\alpha }\frac{\mathrm{\Gamma }(\alpha /2)}{\mathrm{\Gamma }((n-\alpha )/2)}.$

This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ L^p(Rⁿ) with 1 ≤ p < n/α. If p > 1, then the rate of decay of f and that of I_αf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)

$\Vert I_{\alpha }f{\Vert }_{p^{*}}\le C_p\Vert f{\Vert }_p,p^{*}=\frac{np}{n-\alpha p}.$

More generally, the operators I_α are well-defined for complex α such that 0 < Re α < n.

The Riesz potential can be defined more generally in a weak sense as the convolution

$I_{\alpha }f=f*K_{\alpha }$

where K_α is the locally integrable function:

$K_{\alpha }(x)=\frac{1}{c_{\alpha }}\frac{1}{|x{|}^{n-\alpha }}.$

The Riesz potential can therefore be defined whenever f is a compactly supported distribution. In this connection, the Riesz potential of a positive Borel measure μ with compact support is chiefly of interest in potential theory because I_αμ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of Rⁿ.

Consideration of the Fourier transform reveals that the Riesz potential is a Fourier multiplier. In fact, one has

$\widehat{K_{\alpha }}(\xi )=|2\pi \xi {|}^{-\alpha }$

and so, by the convolution theorem,

$\widehat{I_{\alpha }f}(\xi )=|2\pi \xi {|}^{-\alpha }\hat{f}(\xi ).$

The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions

$I_{\alpha }{I}_{\beta }=I_{\alpha +\beta }$

provided

$0<\mathrm{R}\mathrm{e}\alpha ,\mathrm{R}\mathrm{e}\beta <n,0<\mathrm{R}\mathrm{e}(\alpha +\beta )<n.$

Furthermore, if 2 < Re α <n, then

$\mathrm{\Delta }{I}_{\alpha +2}=-I_{\alpha }.$

One also has, for this class of functions,

$\underset{\alpha \to 0^{+}}{\lim }(I^{\alpha }f)(x)=f(x).$

See also

• Bessel potential
• Fractional integration
• Sobolev space
• Fractional Schrödinger equation

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010