Riesz transform

<h2 id="multiplier-properties">Multiplier properties</h2>
The Riesz transforms are given by a <a href="/facts/Fourier_multiplier/ppUR88AH">Fourier multiplier</a>. Indeed, the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of Rjƒ is given by

F
          
        
        (
        
          R
          
            j
          
        
        f
        )
        (
        x
        )
        =
        −
        i
        
          
            
              x
              
                j
              
            
            
              
                |
              
              x
              
                |
              
            
          
        
        (
        
          
            F
          
        
        f
        )
        (
        x
        )
        .
      
    
    {\displaystyle {\mathcal {F}}(R_{j}f)(x)=-i{\frac {x_{j}}{|x|}}({\mathcal {F}}f)(x).}

In this form, the Riesz transforms are seen to be generalizations of the <a href="/facts/Hilbert_transform/RoQaC8MS">Hilbert transform</a>. The kernel is a <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distribution</a> which is <a href="/facts/Homogeneous_function/wF9XX7s5">homogeneous</a> of degree zero. A particular consequence of this last observation is that the Riesz transform defines a <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded linear operator</a> from L2(Rd) to itself.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
This homogeneity property can also be stated more directly without the aid of the Fourier transform. If σs is the <a href="/facts/Homothetic_transformation/WUmuRuzC">dilation</a> on Rd by the scalar s, that is σsx = sx, then σs defines an action on functions via <a href="/facts/Pullback_(differential_geometry)/EzMDBYfK">pullback</a>:

σ
          
            s
          
          
            ∗
          
        
        f
        =
        f
        ∘
        
          σ
          
            s
          
        
        .
      
    
    {\displaystyle \sigma _{s}^{*}f=f\circ \sigma _{s}.}

The Riesz transforms commute with σs:

σ
          
            s
          
          
            ∗
          
        
        (
        
          R
          
            j
          
        
        f
        )
        =
        
          R
          
            j
          
        
        (
        
          σ
          
            x
          
          
            ∗
          
        
        f
        )
        .
      
    
    {\displaystyle \sigma _{s}^{*}(R_{j}f)=R_{j}(\sigma _{x}^{*}f).}

Similarly, the Riesz transforms commute with translations. Let τa be the translation on Rd along the vector a; that is, τa(x) = x + a. Then

τ
          
            a
          
          
            ∗
          
        
        (
        
          R
          
            j
          
        
        f
        )
        =
        
          R
          
            j
          
        
        (
        
          τ
          
            a
          
          
            ∗
          
        
        f
        )
        .
      
    
    {\displaystyle \tau _{a}^{*}(R_{j}f)=R_{j}(\tau _{a}^{*}f).}

For the final property, it is convenient to regard the Riesz transforms as a single <a href="/facts/Vector_(geometric)/bCLrdksy">vectorial</a> entity Rƒ = (R1ƒ,...,Rdƒ). Consider a <a href="/facts/Rotation/yChkz49x">rotation</a> ρ in Rd. The rotation acts on spatial variables, and thus on functions via pullback. But it also can act on the spatial vector Rƒ. The final transformation property asserts that the Riesz transform is <a href="/facts/Equivariant/KkIkDwIH">equivariant</a> with respect to these two actions; that is,

ρ
          
            ∗
          
        
        
          R
          
            j
          
        
        [
        (
        
          ρ
          
            −
            1
          
        
        
          )
          
            ∗
          
        
        f
        ]
        =
        
          ∑
          
            k
            =
            1
          
          
            d
          
        
        
          ρ
          
            j
            k
          
        
        
          R
          
            k
          
        
        f
        .
      
    
    {\displaystyle \rho ^{*}R_{j}[(\rho ^{-1})^{*}f]=\sum _{k=1}^{d}\rho _{jk}R_{k}f.}

These three properties in fact characterize the Riesz transform in the following sense. Let T=(T1,...,Td) be a d-tuple of bounded linear operators from L2(Rd) to L2(Rd) such that

<ul><li>T commutes with all dilations and translations.</li>
<li>T is equivariant with respect to rotations.</li></ul>
Then, for some constant c, T = cR.

<h2 id="relationship-with-the-laplacian">Relationship with the Laplacian</h2>
Somewhat imprecisely, the Riesz transforms of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 give the first <a href="/facts/Partial_derivative/JWHsBkAu">partial derivatives</a> of a solution of the equation

(
          −
          Δ
          
            )
            
              
                1
                2
              
            
          
          u
          =
          f
        
        ,
      
    
    {\displaystyle {(-\Delta )^{\frac {1}{2}}u=f},}

where Δ is the Laplacian. Thus the Riesz transform of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 can be written as:

R
          f
          =
          ∇
          (
          −
          Δ
          
            )
            
              −
              
                
                  1
                  2
                
              
            
          
          f
        
      
    
    {\displaystyle {Rf=\nabla (-\Delta )^{-{\frac {1}{2}}}f}}

In particular, one should also have

R
          
            i
          
        
        
          R
          
            j
          
        
        Δ
        u
        =
        −
        
          
            
              
                ∂
                
                  2
                
              
              u
            
            
              ∂
              
                x
                
                  i
                
              
              ∂
              
                x
                
                  j
                
              
            
          
        
        ,
      
    
    {\displaystyle R_{i}R_{j}\Delta u=-{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}},}

so that the Riesz transforms give a way of recovering information about the entire <a href="/facts/Hessian_matrix/RguMNr3m">Hessian</a> of a function from knowledge of only its Laplacian.
This is now made more precise. Suppose that 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is a <a href="/facts/Schwartz_function/B035faw7">Schwartz function</a>. Then indeed by the explicit form of the Fourier multiplier, one has

R
          
            i
          
        
        
          R
          
            j
          
        
        (
        Δ
        u
        )
        =
        −
        
          
            
              
                ∂
                
                  2
                
              
              u
            
            
              ∂
              
                x
                
                  i
                
              
              ∂
              
                x
                
                  j
                
              
            
          
        
        .
      
    
    {\displaystyle R_{i}R_{j}(\Delta u)=-{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}.}

The identity is not generally true in the sense of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a>. For instance, if 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is a <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">tempered distribution</a> such that 
 
 
 
 Δ
 u
 ∈
 
 L
 
 2
 
 
 (
 
 
 R
 
 
 d
 
 
 )
 
 
 {\displaystyle \Delta u\in L^{2}(\mathbb {R} ^{d})}
 
, then one can only conclude that

∂
                
                  2
                
              
              u
            
            
              ∂
              
                x
                
                  i
                
              
              ∂
              
                x
                
                  j
                
              
            
          
        
        =
        −
        
          R
          
            i
          
        
        
          R
          
            j
          
        
        Δ
        u
        +
        
          P
          
            i
            j
          
        
        (
        x
        )
      
    
    {\displaystyle {\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}=-R_{i}R_{j}\Delta u+P_{ij}(x)}

for some polynomial 
 
 
 
 
 P
 
 i
 j
 
 
 
 
 {\displaystyle P_{ij}}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hilbert_Transform/RoQaC8MS">Hilbert Transform</a></li>
<li><a href="/facts/Poisson_kernel/xkAdJMhu">Poisson kernel</a></li>
<li><a href="/facts/Riesz_potential/dlAMoToY">Riesz potential</a></li></ul>

<ul><li>Gilbarg, D.; <a href="/facts/Neil_Trudinger/4Qsvr1RR">Trudinger, Neil</a> (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-41160-7.</li>
<li><a href="/facts/Elias_Stein/9IRxdGWp">Stein, Elias</a> (1970), Singular integrals and differentiability properties of functions, Princeton University Press.</li>
<li><a href="/facts/Elias_Stein/9IRxdGWp">Stein, Elias</a>; Weiss, Guido (1971), <a href="https://archive.org/details/introductiontofo0000stei">Introduction to Fourier Analysis on Euclidean Spaces</a>, Princeton University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-08078-X.</li>
<li>Arcozzi, N. (1998), Riesz Transform on spheres and compact Lie groups, New York: Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02384766">10.1007/BF02384766</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0004-2080">0004-2080</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119919955">119919955</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Strictly speaking, the definition (1) may only make sense for Schwartz function f. Boundedness on a dense subspace of L2 implies that each Riesz transform admits a continuous linear extension to all of L2. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Riesz transform open-in-new

Riesz transform