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We prove that the coefficientsa   i  j(  x  )${a_{ij}(x)}$   ,  q ( x ) and the domain Ω of a multidimensional fractional diffusion equation can be recovered uniquely from measurementsu  (  b  ,  t  )    ${u(b,t)}$   ,t  ∈  (   t  0   ,   t  1   )    ${t\in (t_0,t_1)}$   , at an arbitrary single point  b  inside a bounded domainΩ  ⊂   ℝ  n${\Omega \subset \mathbb {R}^n}$   . From the measurements we first recover infinitely many spectral data(   λ  m   ,   ϕ  m    (  x  )   )    ${(\lambda _m,\varphi _m(x))}$    of the elliptic operator associated with the fractional diffusion equation. Then, the coefficientsa   i  j(  x  )${a_{ij}(x)}$    and  q ( x ) are found from linear algebraic systems of the formA  y  =  b    ${Ay = b}$   , where  A  is a generalized Wronskian of some set of eigenfunctions that can be shown to be nontrivial. The domain Ω is reconstructed using the first eigenfunctionϕ  1    (  x  )${\varphi _1(x)}$   .

A variable exponent Sobolev theorem for fractional integrals on quasimetric measure spaces