Abstract

In this paper, we propose an iterative shadowgraphic method (ISM) as an interesting alternative to existing methods for self-referencing optical phase retrieval. Two defocused images of the intensity distribution of the light scattered by a weakly absorbing phase object were sufficient to retrieve the transverse phase distribution of the distorted illuminating beam. An algorithm was developed to correct for diffraction effects in phase maps retrieved with a simple shadowgraphic method. We provide a mathematical proof of the convergence of the algorithm to the true profile of the sought phase object. Several numerical tests were performed of the algorithm showing its capability of recovering the full details of the original phase distribution with increased resolution as compared with the simple shadowgraphic method. The convergence of the ISM was also compared numerically with that of a nonoptimized Gerchberg–Saxton-type algorithm and found to be faster and not affected by stagnation problems.

© 2008 Optical Society of America

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References

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  8. A. V. Larichev, P. V. Ivanov, I. G. Iroshnikov, and V. I. Shmal'gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108-1112 (2001).
    [CrossRef]
  9. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express 14, 11919-11924 (2006).
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    [CrossRef]
  27. W. Merzkirch, Flow Visualization (Academic, 1987), pp. 123-124.
  28. D. Pliakis, “On the volume of nodal sets of eigenfunctions,” and “Local estimates for the amplitude growth of waves in inhomogeneous media,” in preparation ; available from the authors: stefano.minardi@uni-jena.de.
  29. A cutoff function is a smooth function that equals one in a closed ball, vanishes outside a larger closed ball, and decays smoothly in the ring.
  30. R. Courant and D. Hilbert, Methods for Mathematical Physics, Vol. I, II (Wiley Interscience, 1989).
  31. We define as approximate an equation F with respect to another equation H if we can find two constants c1 and c2 such that c1H<F<c2H.
  32. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).
  33. M. Giglio, D. Borgiol, M. A. C. Potenza, and A. Valiati, “Near field scattering,” Phys. Chem. Chem. Phys. 6, 1547-1550 (2004).
    [CrossRef]
  34. C. Cohen-Tannoudji, B. Diu, and F. Laoe, Quantum Mechanics (Wiley-Interscience, 2006).
  35. P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intanstätverteilung in Spectrallinien II,” Z. Phys. 69, 298-308 (1931).
    [CrossRef]
  36. J. L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach (Cambridge U. Press, 1998).
    [CrossRef]
  37. D. Pliakis, “On a generalized Hardy's inequality and its applications in spectral geometry,” arXiv-math.AP:0203092.

2008 (2)

S. Minardi, A. Gopal, M. Tatarakis, A. Couairon, G. Tamosauskas, R. Piskarskas, A. Dubietis, and P. Di Trapani, “Time-resolved refractive index and absorption mapping of light plasma filaments in water,” Opt. Lett. 33, 86-88 (2008).
[CrossRef]

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

2007 (2)

2006 (1)

2004 (1)

M. Giglio, D. Borgiol, M. A. C. Potenza, and A. Valiati, “Near field scattering,” Phys. Chem. Chem. Phys. 6, 1547-1550 (2004).
[CrossRef]

2003 (3)

2002 (2)

S. P. Trainoff and D. S. Cannell, “Physical optics treatment of shadowgraphy,” Phys. Fluids 14, 1340-1363 (2002).
[CrossRef]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O'Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

2001 (2)

L. J. Allen and M. P. Oxley, “Phase retrieval for series of images obtained by defocus variation,” Opt. Commun. 199, 65-75 (2001).
[CrossRef]

A. V. Larichev, P. V. Ivanov, I. G. Iroshnikov, and V. I. Shmal'gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108-1112 (2001).
[CrossRef]

2000 (1)

1998 (1)

1995 (1)

1994 (1)

1993 (2)

1982 (3)

1979 (1)

1974 (1)

J. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astrophys. J., Suppl. 15, 417-426 (1974).

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

1931 (1)

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intanstätverteilung in Spectrallinien II,” Z. Phys. 69, 298-308 (1931).
[CrossRef]

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval for series of images obtained by defocus variation,” Opt. Commun. 199, 65-75 (2001).
[CrossRef]

Allman, B. E.

Barty, A.

Bijaoui, A.

J. L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach (Cambridge U. Press, 1998).
[CrossRef]

Borgiol, D.

M. Giglio, D. Borgiol, M. A. C. Potenza, and A. Valiati, “Near field scattering,” Phys. Chem. Chem. Phys. 6, 1547-1550 (2004).
[CrossRef]

Borgsmüller, S.

Brueck, S. R. J.

Cannell, D. S.

S. P. Trainoff and D. S. Cannell, “Physical optics treatment of shadowgraphy,” Phys. Fluids 14, 1340-1363 (2002).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laoe, Quantum Mechanics (Wiley-Interscience, 2006).

Couairon, A.

Courant, R.

R. Courant and D. Hilbert, Methods for Mathematical Physics, Vol. I, II (Wiley Interscience, 1989).

Dannberg, P.

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Di Trapani, P.

Dietrich, C.

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laoe, Quantum Mechanics (Wiley-Interscience, 2006).

Dubietis, A.

Fienup, J. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

Genoud, G.

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Georgiadou, E.

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Giglio, M.

M. Giglio, D. Borgiol, M. A. C. Potenza, and A. Valiati, “Near field scattering,” Phys. Chem. Chem. Phys. 6, 1547-1550 (2004).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Gopal, A.

Guilbaud, O.

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Gureyev, T. E.

Hilbert, D.

R. Courant and D. Hilbert, Methods for Mathematical Physics, Vol. I, II (Wiley Interscience, 1989).

Hodgson, K. O.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O'Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Högbom, J.

J. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astrophys. J., Suppl. 15, 417-426 (1974).

Iroshnikov, I. G.

A. V. Larichev, P. V. Ivanov, I. G. Iroshnikov, and V. I. Shmal'gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108-1112 (2001).
[CrossRef]

Ivanov, P. V.

A. V. Larichev, P. V. Ivanov, I. G. Iroshnikov, and V. I. Shmal'gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108-1112 (2001).
[CrossRef]

Jüptner, W.

Kämpfe, T.

Keen, S.

Kley, E.-B.

Kresse, T.

Kuznestova, Y.

Laoe, F.

C. Cohen-Tannoudji, B. Diu, and F. Laoe, Quantum Mechanics (Wiley-Interscience, 2006).

Larichev, A. V.

A. V. Larichev, P. V. Ivanov, I. G. Iroshnikov, and V. I. Shmal'gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108-1112 (2001).
[CrossRef]

Leach, J.

L'Hullier, A.

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Love, G. D.

Männer, R.

Marchesini, S.

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef] [PubMed]

Marcuse, D.

McMahon, P. J.

Mengotti, E.

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Merzkirch, W.

W. Merzkirch, Flow Visualization (Academic, 1987), pp. 123-124.

Miao, J.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O'Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Minardi, S.

Murtagh, F.

J. L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach (Cambridge U. Press, 1998).
[CrossRef]

Noethe, S.

Nugent, K. A.

Ohsuna, T.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O'Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

O'Keefe, M. A.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O'Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval for series of images obtained by defocus variation,” Opt. Commun. 199, 65-75 (2001).
[CrossRef]

Padgett, M. J.

Paganin, D.

Pettersson, S.-G.

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Piskarskas, R.

Pliakis, D.

D. Pliakis, “On a generalized Hardy's inequality and its applications in spectral geometry,” arXiv-math.AP:0203092.

D. Pliakis, “On the volume of nodal sets of eigenfunctions,” and “Local estimates for the amplitude growth of waves in inhomogeneous media,” in preparation ; available from the authors: stefano.minardi@uni-jena.de.

Potenza, M. A. C.

M. Giglio, D. Borgiol, M. A. C. Potenza, and A. Valiati, “Near field scattering,” Phys. Chem. Chem. Phys. 6, 1547-1550 (2004).
[CrossRef]

Pourtal, E.

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

Roberts, A.

Roddier, C.

Roddier, F.

Saunter, C.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Schnars, U.

Schwarz, C. J.

Shmal'gauzen, V. I.

A. V. Larichev, P. V. Ivanov, I. G. Iroshnikov, and V. I. Shmal'gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108-1112 (2001).
[CrossRef]

Starck, J. L.

J. L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach (Cambridge U. Press, 1998).
[CrossRef]

Tamosauskas, G.

Tatarakis, M.

Teague, M. R.

Terasaki, O.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O'Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

Tiller, J. B.

Trainoff, S. P.

S. P. Trainoff and D. S. Cannell, “Physical optics treatment of shadowgraphy,” Phys. Fluids 14, 1340-1363 (2002).
[CrossRef]

Tünnermann, A.

Valiati, A.

M. Giglio, D. Borgiol, M. A. C. Potenza, and A. Valiati, “Near field scattering,” Phys. Chem. Chem. Phys. 6, 1547-1550 (2004).
[CrossRef]

Van Cittert, P. H.

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intanstätverteilung in Spectrallinien II,” Z. Phys. 69, 298-308 (1931).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

Wahlström, G.-C.

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. B (1)

G. Genoud, O. Guilbaud, E. Mengotti, S.-G. Pettersson, E. Georgiadou, E. Pourtal, G.-C. Wahlström, and A. L'Hullier, “XUV digital in-line holography using higher order harmonics,” Appl. Phys. B 90, 533-538 (2008).
[CrossRef]

Astrophys. J., Suppl. (1)

J. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astrophys. J., Suppl. 15, 417-426 (1974).

in preparation (1)

D. Pliakis, “On the volume of nodal sets of eigenfunctions,” and “Local estimates for the amplitude growth of waves in inhomogeneous media,” in preparation ; available from the authors: stefano.minardi@uni-jena.de.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

L. J. Allen and M. P. Oxley, “Phase retrieval for series of images obtained by defocus variation,” Opt. Commun. 199, 65-75 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Optik (Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Phys. Chem. Chem. Phys. (1)

M. Giglio, D. Borgiol, M. A. C. Potenza, and A. Valiati, “Near field scattering,” Phys. Chem. Chem. Phys. 6, 1547-1550 (2004).
[CrossRef]

Phys. Fluids (1)

S. P. Trainoff and D. S. Cannell, “Physical optics treatment of shadowgraphy,” Phys. Fluids 14, 1340-1363 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, and M. A. O'Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Quantum Electron. (1)

A. V. Larichev, P. V. Ivanov, I. G. Iroshnikov, and V. I. Shmal'gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108-1112 (2001).
[CrossRef]

Rev. Sci. Instrum. (1)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[CrossRef] [PubMed]

Ultrason. Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Z. Phys. (1)

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intanstätverteilung in Spectrallinien II,” Z. Phys. 69, 298-308 (1931).
[CrossRef]

Other (10)

J. L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach (Cambridge U. Press, 1998).
[CrossRef]

D. Pliakis, “On a generalized Hardy's inequality and its applications in spectral geometry,” arXiv-math.AP:0203092.

C. Cohen-Tannoudji, B. Diu, and F. Laoe, Quantum Mechanics (Wiley-Interscience, 2006).

A cutoff function is a smooth function that equals one in a closed ball, vanishes outside a larger closed ball, and decays smoothly in the ring.

R. Courant and D. Hilbert, Methods for Mathematical Physics, Vol. I, II (Wiley Interscience, 1989).

We define as approximate an equation F with respect to another equation H if we can find two constants c1 and c2 such that c1H<F<c2H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

W. Merzkirch, Flow Visualization (Academic, 1987), pp. 123-124.

For a historical introduction of Shack-Hartman sensors see http://www.wavefrontsciences.com/Historical%20Development.pdf.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

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Figures (5)

Fig. 1
Fig. 1

Block diagram of the iterative shadowgraphic method.

Fig. 2
Fig. 2

Retrieval of a phase profile by means of the ISM algorithm. (a) Phase profile of the object (portrait of Christiaan Huygens). (b) Retrieved phase profile after 500 iterations. (c) Intensity field of the object (portrait of Isaac Newton, I O ). (d) Intensity field after a propagation of z = 11.2 z 0 ( I S ) . Frames (c) and (d) are the input data for the ISM algorithm. Input conditions: intensity modulation depth = 10 % ; phase modulation amplitude = 0.05 rad . The coordinates of the images are expressed in pixel units.

Fig. 3
Fig. 3

Accuracy of the phase retrieval as a function of the spatial frequency. (a) Amplitude of the spectrum of the retrieved phase field normalized to the spectral amplitude of the phase profile of the object. (b) Unwrapped phase difference between the spectrum of the retrieved phase field and the spectral phase of the phase profile of the object. In both graphs, the results after the first iteration are plotted as dashed curves lines, while data after 500 iterations are displayed as a continuous curve.

Fig. 4
Fig. 4

Comparison between the convergence rate of the ISM (continuous curve) and of the iterative propagation method (dashed curve). The standard deviation of the difference between the estimated and the true phase profile is calculated for each iteration step.

Fig. 5
Fig. 5

Comparison of the resolution of the retrieval for the shadowgraphic, the ISM, and the TIE method. The intensity of the object was again the picture of Newton, while the phase was chosen to map the USAF 1951 resolution test. Phase retrieval with (a) the simple shadowgraphic method, (b) the ISM method after 500 iterations, and (c) the TIE one. The distance between the images used by the ISM method is 2.5 times longer than the maximum distance between the sampled planes used for the TIE algorithm.

Equations (38)

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2 ϕ 0 ( x , y ) = k Δ z [ 1 I S ( x , y ) I O ( x , y ) ] .
U n = I O e i Ψ n .
I n = F Δ z U n 2 ,
F Δ z ( x ) = k e i k Δ z i 2 π Δ z e i k Δ z ( x 2 + y 2 )
2 ϕ n = k Δ z ( 1 I n I S ) .
Ψ n + 1 = Ψ n + α n ϕ n ,
α n = ( 1 ) n α
I n vol ( Ω ) k ( 2 π Δ z ) [ Ω I 0 ( x , y , Δ z ) d x d y ] ,
I n I m C k Δ z I 0 [ Ω Ψ n Ψ m 2 d x d y ] 1 2 ,
1 σ 2 Ψ n + E I n I O 1 σ 2 Ψ n + E + ,
ζ n = w n ϕ n .
K n , m = ϕ n ζ m n , m N .
K n , m ϕ n ϕ m 1 w m ϕ m .
ϵ n , m = 1 I n I S w m ( 1 I m I S ) ,
η m = c m w m .
I n I m j C diam ( Ω N j ) [ Ω N j ( Ψ n Ψ m ) 2 ] 1 2 C j diam ( Ω N j ) σ ( Ω N j I n I m 2 I 0 2 d x d y ) 1 2 ,
2 K n , m ϵ n , m + 2 ( ζ m H m 0 + ζ m H m 1 ζ m H m 1 2 ) ,
H n 0 = 2 η n η n , H n 1 = η n η n .
Ω χ 2 K n ϵ + Ω χ ζ m H m 0 + χ ζ m H m 1 + χ ζ m H m 1 2 ,
Ω χ 2 K n ϵ + γ 1 Ω χ ζ m ( H m 0 + H m 1 2 ) + γ 2 ζ m χ H m 1 .
Ω u 2 C Ω u 2 .
( Ω u 2 s l ) 1 2 s C s [ vol ( Ω ) ] 2 s + 3 2 s + 2 ( Ω u 2 l 2 u 2 ) 1 2 .
Ω φ u 2 l 1 ( 2 u )
Ω χ K n , m 2 l 1 2 K n C 1 Ω η m K n , m 2 l 1 ( χ ζ m ) χ ζ m 2 .
( Ω χ K n , m 2 s l ) 1 2 s C s [ vol ( Ω ) ] 2 n + 3 2 s + 2 ϵ Ω K n , m 2 l 1 ( χ ζ m ) χ ζ m 2 ,
lim r [ 1 vol ( Ω ) Ω f r ] 1 r = sup Ω f ,
sup Ω K n , m C ϵ ,
sup ϕ n 0 .
P ( x 1 , x 2 ) = i = 1 N [ x 2 g j ( x 1 1 l j ) ] ,
P P 2 2 j = 1 N 1 [ x 2 g j ( x 1 ) ] 2 [ 1 + g 2 x 1 2 ( 1 l j 1 ) ] .
supp ( χ 1 ) V ϵ ( P ) = { x R 2 d [ x , V ( P ) ] < ϵ } ,
R f 2 x 2 4 R f 2 ,
R 2 P P 2 f 2 C R 2 f 2 + f 2 + ϵ ,
χ ( x ) = ϕ [ P ( x ) σ ] ϕ [ σ 2 P ( x ) ] ,
R 2 P P 2 f 2 c R 2 f 2 + f 2 .
Ω χ ζ P P 2 C Ω P χ ζ 2 2 C + ϵ 8 Ω χ ζ P P 2 + 2 C + 1 8 ϵ Ω P ( χ ζ ) χ ζ 2 ,
Ω χ ζ P P 2 C Ω P ( χ ζ ) χ ζ 2 .
Ω χ ζ 2 P P C Ω P ( χ ζ ) χ ζ 2 .

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